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PDF Editor FAQ
Can a solution [math] (x,y,z,t) [/math] in positive co-prime integers (pairwise) be found for [math] x^3+y^3+z^3=t^3 [/math]?
There is one well-known solution for this problem [math](3, 4,5,6)[/math]:[math]3^3 +4^3 +5^3=6^3[/math] or [math](1,6,8,9)[/math] (Update: Well, they don't satisfy a rather artificial pairwise gcd condition).To describe all solutions we can make use of the underlying geometry. This equation describes the famous cubic Fermat surface [math]C[/math] in [math]\mathbf{P}^3[/math].All [math]27[/math] lines on [math]C[/math] are particularly easy to describe.Although Fermat surface has no rational pairs of skew lines, yet it still haspairs of complex conjugate ones; for example, you can choose one line in [math]\mathbf{P}^3[/math] given by [math]L: x+\varepsilon y=0, t-\varepsilon z=0[/math] and a second line given by [math]M: x+\overline{\varepsilon} y=0, t-\overline{\varepsilon} z=0.[/math] Here [math]\varepsilon[/math] is a primitive 3rd root of unity.This condition is sufficient (in fact, necessary as well) for existence of rational parametrization; check Page on arxiv.org §3 for a simple geometrical construction how to obtain it. The idea pertains to construction of a rational plane that contains a rational line which intersects both skew lines. A point of this rational planes uniquely defines a point on [math]C[/math] by constructing the unique line through this point meeting both skew lines. I this way one gets a rational parametrization [math]\mathbf{P}^2 \to C[/math].The result is not that elegant as the above construction though.Another parametrization Complete cubic parametrization of w^3+x^3+y^3+z^3=0 uses the fact that Fermat surface (in general, a smooth cubic surface) is a blow-up of [math]\mathbf{P}^2[/math] at [math]6[/math] points. In fact, parametrization is the inverse construction (i.e. blow-down at 6 skew lines).And finally the most old and interesting rational parametrization of [math]C[/math] was found by Leonhard Euler. It makes use of polynomials of degree [math] 4[/math] unlike two former examples, but the formula looks nice and seem to be well adjusted for search of positive quadruples:[math] \begin{matrix} x=c^4 -c(a-3b)(a^2+3b^2) \\ y=c(a+3b)(a^2 +3b^2) -c^4\\ z=(a^2 +3b^2)^2 -c^3(a+3b)\\ t=(a^2 +3b^2)^2 +(3b-a)c^3\end{matrix}[/math](cf: History of the Theory of Numbers, vol.2., pp 552-554 )However, since Euler's parametrization of degree 4 it one should probably work with rational numbers to get satisfactory (pairwise co-prime) results.The parametrization of Elkies works for this purpose much better producing loads of counterexamples:[math] \begin{array}{ll} x=-a^3-2a^2 c +3 a^2b +12abc \\ -3ab^2-4ac^2+6b^2c +12bc^2 +9b^3\\ ~\\ y=a^3+2a^2c +3a^2b +12abc +\\ 3ab^2 + 4ac^2-6b^2c +12bc^2 +9b^3\\ ~\\ z=-8c^3-8ac^2-9b^3-a^3\\ -3a^2b-3ab^2-4a^2c -12b^2c\\ ~\\ t=-8c^3-8ac^2+9b^3-a^3+3a^2b\\ -3ab^2-4a^2c-12b^2c \end{array}[/math]The following Python code implements brute force search of counterexamples :def gcdmod(l): #gcd of numbers in the list from fractions import gcd; a=l[0] for x in l: a=gcd(a,x) return(a) def test(l): # checks whether a guadruple admissible (pairwise prime and positive) from fractions import gcd; from itertools import combinations; for x in l: if (x<=0): return False y=fcdmod(l); ld=[] for x in l: ld.append(x/y) for k in combinations(ld,2): if (gcd(k[0],k[1])>1): return False return(True) def generator(n): #generates quadruples and picks counterexamples out for c in range(-n,n+1): for a in range(-n,n+1): for b in range(-n,n+1): x=-a**3-2*(a**2)*c +3*(a**2)*b +12*a*b*c -3*a*(b**2)-4*a*(c**2)+6*b*b*c +12*b*(c**2) +9*(b**3); y=a**3+2*(a**2)*c +3*(a**2)*b +12*a*b*c +3*a*(b**2) + 4*a*(c**2)-6*b*b*c +12*b*(c**2)+9*(b**3); z=-8*c**3-8*a*(c**2)-9*b**3-a**3-3*(a**2)*b-3*a*(b**2)-4*(a**2)*c -12*(b**2)*c; t=-(8*c**3+8*a*(c**2)-9*b**3+a**3-3*(a**2)*b+3*a*(b**2)+4*(a**2)*c+12*(b**2)*c); res=x**3+y**3+z**3-t**3; if test([x,y,z,t]): m=gcdmod([x,y,z,t]); print("(",x//m,",",y//m,",",z//m,",",t//m,")"); Results (counterexamples/solutions)>>> generator(20) ( 3989 , 877 , 5983 , 6529 ) ( 41779 , 19043 , 49877 , 58859 ) ( 65077 , 49253 , 54227 , 82157 ) ( 44923 , 32861 , 46619 , 61043 ) ( 69229 , 66887 , 50033 , 91109 ) ( 14311 , 851 , 35429 , 36191 ) ( 29257 , 20117 , 39203 , 45377 ) ( 4733 , 4063 , 5197 , 6793 ) ( 383 , 379 , 349 , 535 ) ( 31441 , 31943 , 36977 , 48521 ) ( 51067 , 63449 , 40031 , 76787 ) ( 1667 , 1399 , 3541 , 3727 ) ( 19063 , 20159 , 30521 , 35183 ) ( 33433 , 44897 , 34583 , 55313 ) ( 3019 , 5807 , 19913 , 20099 ) ( 1931 , 3697 , 3163 , 4471 ) ( 21319 , 42251 , 26669 , 47039 ) ( 4003 , 11993 , 18527 , 20123 ) ( 11029 , 28739 , 21941 , 32909 ) ( 22591 , 56501 , 22979 , 58871 ) ( 109 , 437 , 293 , 479 ) ( 11101 , 39869 , 19451 , 41621 ) ( 2099 , 6883 , 2377 , 7039 ) ( 2347 , 38183 , 12497 , 38627 ) ( 1699 , 52337 , 6983 , 52379 ) ( 48365 , 16561 , 52333 , 63907 ) ( 31523 , 6073 , 44581 , 49345 ) ( 78623 , 65269 , 51049 , 96445 ) ( 34715 , 18331 , 40987 , 48889 ) ( 55421 , 45949 , 44833 , 70963 ) ( 37379 , 30613 , 38281 , 51553 ) ( 59237 , 62191 , 40903 , 80251 ) ( 10991 , 1867 , 28267 , 28813 ) ( 39947 , 43783 , 35599 , 57769 ) ( 701 , 535 , 1363 , 1447 ) ( 25385 , 29359 , 29959 , 40927 ) ( 42851 , 58705 , 32077 , 67969 ) ( 5573 , 4189 , 20929 , 21115 ) ( 15527 , 27697 , 22957 , 33349 ) ( 29033 , 55243 , 24139 , 59167 ) ( 1427 , 5671 , 15343 , 15601 ) ( 16271 , 38659 , 20659 , 41389 ) ( 1931 , 10897 , 14221 , 16105 ) ( 7373 , 26059 , 16843 , 28387 ) ( 17351 , 52237 , 16657 , 53413 ) ( 1475 , 17011 , 13123 , 19297 ) ( 491 , 24877 , 11185 , 25609 ) ( 7373 , 50119 , 9199 , 50275 ) ( 13123 , 1475 , 17011 , 19297 ) ( 135665 , 46801 , 140291 , 175013 ) ( 157979 , 72955 , 143513 , 193871 ) ( 182285 , 101869 , 146567 , 217313 ) ( 238601 , 169705 , 150443 , 279677 ) ( 271475 , 209491 , 150401 , 319463 ) ( 99037 , 13751 , 130141 , 147025 ) ( 118423 , 35993 , 132955 , 159499 ) ( 161587 , 87317 , 139519 , 196615 ) ( 186229 , 117263 , 142405 , 222121 ) ( 213487 , 150641 , 144451 , 252835 ) ( 9371 , 599 , 13693 , 15023 ) ( 1627 , 415 , 1997 , 2311 ) ( 1555 , 517 , 1769 , 2113 ) ( 15833 , 8213 , 14719 , 19757 ) ( 18355 , 11311 , 15077 , 22279 ) ( 2831 , 2081 , 1933 , 3389 ) ( 24239 , 18635 , 15529 , 29027 ) ( 27697 , 22957 , 15527 , 33349 ) ( 3697 , 3163 , 1931 , 4471 ) ( 87803 , 17209 , 118907 , 133199 ) ( 168851 , 116497 , 131939 , 205607 ) ( 27779 , 21277 , 19151 , 33647 ) ( 255089 , 225991 , 133901 , 312581 ) ( 74233 , 9137 , 112291 , 122221 ) ( 10313 , 2341 , 14201 , 15839 ) ( 108757 , 50381 , 118567 , 145513 ) ( 14771 , 7759 , 15047 , 19217 ) ( 127999 , 74423 , 122173 , 163027 ) ( 2867 , 2071 , 2267 , 3533 ) ( 172603 , 131507 , 128161 , 211951 ) ( 198829 , 165413 , 129679 , 244225 ) ( 12013 , 10709 , 6823 , 14857 ) ( 2867 , 745 , 4019 , 4463 ) ( 4853 , 3235 , 4409 , 6161 ) ( 7183 , 461 , 12625 , 13357 ) ( 11041 , 5111 , 13375 , 15703 ) ( 96659 , 51463 , 108845 , 132551 ) ( 114437 , 74521 , 112571 , 149465 ) ( 15499 , 10889 , 14293 , 19945 ) ( 134063 , 100483 , 115985 , 170819 ) ( 20989 , 18227 , 14947 , 26515 ) ( 180587 , 162847 , 120149 , 228575 ) ( 208349 , 200113 , 120035 , 265841 ) ( 1723 , 173 , 3103 , 3271 ) ( 67807 , 22517 , 98911 , 108883 ) ( 82945 , 41519 , 102373 , 119701 ) ( 99259 , 62801 , 106147 , 134287 ) ( 117181 , 86795 , 109801 , 153073 ) ( 159577 , 144647 , 115021 , 204973 ) ( 213589 , 218531 , 114577 , 278857 ) ( 293 , 109 , 437 , 479 ) ( 3163 , 1931 , 3697 , 4471 ) ( 5197 , 4733 , 4063 , 6793 ) ( 45827 , 8341 , 87167 , 91223 ) ( 58949 , 24463 , 90053 , 98297 ) ( 72767 , 42481 , 93635 , 108659 ) ( 87713 , 62827 , 97481 , 122741 ) ( 104219 , 85933 , 101159 , 140975 ) ( 17531 , 16033 , 14891 , 23399 ) ( 143639 , 142153 , 106283 , 191627 ) ( 194483 , 214597 , 105551 , 264071 ) ( 61261 , 33785 , 87643 , 98017 ) ( 8495 , 5353 , 11189 , 12941 ) ( 10705 , 7493 , 13063 , 15709 ) ( 1751 , 1525 , 1733 , 2417 ) ( 106675 , 97919 , 98677 , 146023 ) ( 4321 , 1207 , 8765 , 9109 ) ( 209 , 107 , 337 , 365 ) ( 7037 , 4787 , 9505 , 10961 ) ( 379 , 349 , 383 , 535 ) ( 1733 , 1751 , 1525 , 2417 ) ( 14293 , 15499 , 10889 , 19945 ) ( 5869 , 3329 , 9757 , 10531 ) ( 52811 , 34675 , 79841 , 88703 ) ( 65213 , 52453 , 83663 , 100241 ) ( 691 , 599 , 823 , 1033 ) ( 94265 , 96289 , 90803 , 135341 ) ( 12841 , 13661 , 11521 , 18367 ) ( 111779 , 123211 , 93257 , 159767 ) ( 131861 , 154045 , 94391 , 189353 ) ( 154943 , 189223 , 93773 , 224531 ) ( 32551 , 12977 , 71299 , 73627 ) ( 43273 , 27179 , 74377 , 80029 ) ( 66949 , 61991 , 81901 , 102841 ) ( 114523 , 136709 , 90127 , 168559 ) ( 159319 , 206561 , 88531 , 238411 ) ( 911 , 283 , 2465 , 2507 ) ( 2081 , 1933 , 2831 , 3389 ) ( 349 , 383 , 379 , 535 ) ( 3659 , 4471 , 3173 , 5543 ) ( 17291 , 3025 , 62243 , 62687 ) ( 26765 , 14731 , 64361 , 66113 ) ( 36359 , 28045 , 67463 , 72251 ) ( 46505 , 43399 , 71117 , 81533 ) ( 70181 , 81955 , 78353 , 111257 ) ( 84575 , 106021 , 81071 , 132563 ) ( 101249 , 133855 , 82613 , 158741 ) ( 1889 , 529 , 7373 , 7415 ) ( 19603 , 9683 , 59905 , 60679 ) ( 4139 , 3547 , 8027 , 8585 ) ( 47809 , 51713 , 69715 , 83701 ) ( 58987 , 70571 , 73321 , 98335 ) ( 5521 , 7121 , 5875 , 9013 ) ( 1409 , 1873 , 1385 , 2291 ) ( 103897 , 147737 , 79099 , 169165 ) ( 124099 , 181763 , 77905 , 203191 ) ( 1111 , 1061 , 2263 , 2419 ) ( 319 , 425 , 379 , 547 ) ( 6983 , 1699 , 52337 , 52379 ) ( 2377 , 2099 , 6883 , 7039 ) ( 22979 , 22591 , 56501 , 58871 ) ( 40031 , 51067 , 63449 , 76787 ) ( 50033 , 69229 , 66887 , 91109 ) ( 75149 , 115561 , 71531 , 132641 ) ( 10351 , 16193 , 8989 , 18253 ) ( 15073 , 24587 , 8587 , 26647 ) ( 9199 , 7373 , 50119 , 50275 ) ( 16657 , 17351 , 52237 , 53413 ) ( 24139 , 29033 , 55243 , 59167 ) ( 32077 , 42851 , 58705 , 67969 ) ( 40903 , 59237 , 62191 , 80251 ) ( 51049 , 78623 , 65269 , 96445 ) ( 77029 , 128123 , 68473 , 142297 ) ( 421 , 445 , 5207 , 5209 ) ( 103 , 115 , 659 , 661 ) ( 5717 , 9127 , 44909 , 45065 ) ( 12239 , 18313 , 46955 , 48131 ) ( 18737 , 29251 , 49841 , 53765 ) ( 25643 , 42373 , 53135 , 62399 ) ( 42407 , 76897 , 59219 , 90395 ) ( 907 , 6047 , 41941 , 41983 ) ( 7165 , 14129 , 43363 , 43921 ) ( 1271 , 2341 , 5561 , 5717 ) ( 13159 , 23771 , 45817 , 48187 ) ( 19321 , 35405 , 48871 , 55213 ) ( 2825 , 5263 , 6311 , 7487 ) ( 26083 , 49463 , 52093 , 65431 ) ( 33877 , 66377 , 55051 , 79273 ) ( 4787 , 9505 , 7037 , 10961 ) ( 43135 , 86579 , 57313 , 97171 ) ( 54289 , 110501 , 58447 , 119557 ) ( 7589 , 15499 , 7307 , 16571 ) ( 84013 , 171233 , 55603 , 179521 ) ( 11663 , 23677 , 6689 , 24749 ) ( 509 , 1087 , 1661 , 1817 ) ( 1423 , 3671 , 5311 , 5869 ) ( 14039 , 35143 , 43949 , 50795 ) ( 2831 , 6943 , 6701 , 8699 ) ( 34757 , 84661 , 51455 , 92249 ) ( 4939 , 11987 , 6499 , 12841 ) ( 44783 , 108127 , 52181 , 114371 ) ( 10301 , 24013 , 6911 , 24809 ) ( 4435 , 19781 , 37711 , 39463 ) ( 9109 , 29375 , 40213 , 45001 ) ( 26731 , 73565 , 47863 , 80767 ) ( 5035 , 13553 , 7003 , 14359 ) ( 19 , 599 , 1285 , 1327 ) ( 517 , 1769 , 1555 , 2113 ) ( 107 , 337 , 209 , 365 ) ( 4787 , 29377 , 35891 , 41543 ) ( 8885 , 40699 , 38393 , 49961 ) ( 13679 , 54781 , 40727 , 61667 ) ( 19601 , 72055 , 42461 , 77093 ) ( 36557 , 117907 , 42401 , 120833 ) ( 63209 , 181711 , 34757 , 184637 ) ( 1009 , 24761 , 32731 , 36901 ) ( 341 , 3673 , 4229 , 5003 ) ( 8461 , 46997 , 37279 , 53857 ) ( 1343 , 6787 , 4787 , 7517 ) ( 13303 , 62399 , 39157 , 67339 ) ( 27379 , 103931 , 39961 , 106471 ) ( 50191 , 162647 , 33949 , 164707 ) ( 443 , 4469 , 3775 , 5231 ) ( 877 , 5983 , 3989 , 6529 ) ( 2153 , 10139 , 4153 , 10397 ) ( 3091 , 12877 , 4007 , 13063 ) ( 5807 , 19913 , 3019 , 20099 ) ( 23 , 4915 , 4409 , 5891 ) ( 205 , 5003 , 3979 , 5731 ) ( 3275 , 46351 , 32741 , 51263 ) ( 7229 , 61345 , 34163 , 64721 ) ( 1207 , 8765 , 4321 , 9109 ) ( 2953 , 14375 , 4111 , 14527 ) ( 28451 , 128935 , 31373 , 130007 ) ( 40085 , 160441 , 26651 , 161513 ) ( 12079 , 89813 , 31615 , 91171 ) ( 2791 , 16313 , 4243 , 16435 ) ( 437 , 3733 , 1031 , 3761 ) ( 115 , 659 , 103 , 661 ) ( 5141 , 88687 , 25829 , 89417 ) ( 11663 , 112849 , 23267 , 113219 ) ( 31979 , 175501 , 11591 , 175871 ) ( 4573 , 99737 , 21691 , 100081 ) ( 983 , 14041 , 2501 , 14069 ) ( 11719 , 126083 , 17809 , 126235 ) ( 1123 , 8279 , 613 , 8287 ) ( 3401 , 21859 , 947 , 21887 ) ( 445 , 5207 , 421 , 5209 ) ( 3917 , 125317 , 10799 , 125345 ) ( 991 , 17531 , 955 , 17533 ) ( 12599 , 156463 , 3845 , 156491 ) ( 3925 , 140039 , 3853 , 140041 ) ( 30661 , 4343 , 38531 , 44159 ) ( 76897 , 59219 , 42407 , 90395 ) ( 33661 , 15713 , 34913 , 43895 ) ( 20803 , 7001 , 28937 , 32261 ) ( 10993 , 785 , 23729 , 24491 ) ( 13129 , 8339 , 21479 , 23363 ) ( 24835 , 25877 , 24749 , 36293 ) ( 42373 , 53135 , 25643 , 62399 ) ( 2071 , 2267 , 2867 , 3533 ) ( 26779 , 36791 , 22403 , 43133 ) ( 15529 , 24239 , 18635 , 29027 ) ( 29251 , 49841 , 18737 , 53765 ) ( 7759 , 15047 , 14771 , 19217 ) ( 16657 , 34313 , 16313 , 36683 ) ( 2605 , 8783 , 11243 , 12839 ) ( 7975 , 22577 , 13361 , 24329 ) ( 18313 , 46955 , 12239 , 48131 ) ( 2341 , 14201 , 10313 , 15839 ) ( 8287 , 32243 , 10631 , 32801 ) ( 1741 , 21323 , 8495 , 21767 ) ( 9127 , 44909 , 5717 , 45065 ) ( 3775 , 443 , 4469 , 5231 ) ( 39311 , 13735 , 36793 , 48367 ) ( 9505 , 7037 , 4787 , 10961 ) ( 42359 , 25633 , 33343 , 50647 ) ( 66377 , 55051 , 33877 , 79273 ) ( 4471 , 3173 , 3659 , 5543 ) ( 45599 , 38131 , 30061 , 56191 ) ( 16475 , 7009 , 22783 , 25531 ) ( 29837 , 24907 , 25621 , 38893 ) ( 49463 , 52093 , 26083 , 65431 ) ( 8249 , 1513 , 18295 , 18841 ) ( 1621 , 941 , 2435 , 2693 ) ( 18227 , 14947 , 20989 , 26515 ) ( 383 , 379 , 349 , 535 ) ( 5263 , 6311 , 2825 , 7487 ) ( 1415 , 1117 , 2371 , 2599 ) ( 35405 , 48871 , 19321 , 55213 ) ( 4007 , 3091 , 12877 , 13063 ) ( 10889 , 14293 , 15499 , 19945 ) ( 1933 , 2831 , 2081 , 3389 ) ( 21443 , 33487 , 17041 , 37411 ) ( 11633 , 21799 , 14137 , 24529 ) ( 23771 , 45817 , 13159 , 48187 ) ( 893 , 3973 , 8923 , 9181 ) ( 355 , 983 , 1241 , 1427 ) ( 5111 , 13375 , 11041 , 15703 ) ( 12569 , 31201 , 11647 , 32377 ) ( 2341 , 5561 , 1271 , 5717 ) ( 1013 , 7783 , 8185 , 10069 ) ( 5087 , 20281 , 9511 , 21055 ) ( 14129 , 43363 , 7165 , 43921 ) ( 461 , 12625 , 7183 , 13357 ) ( 283 , 2465 , 911 , 2507 ) ( 6047 , 41941 , 907 , 41983 ) ( 607 , 11 , 751 , 865 ) ( 92117 , 19057 , 95273 , 118247 ) ( 109631 , 38491 , 97079 , 131801 ) ( 18407 , 8587 , 14123 , 21317 ) ( 78763 , 11561 , 89809 , 106693 ) ( 112519 , 49541 , 93589 , 133249 ) ( 178453 , 125771 , 96199 , 204583 ) ( 237649 , 194471 , 90859 , 278179 ) ( 84011 , 31237 , 83093 , 106181 ) ( 100253 , 50287 , 85283 , 120623 ) ( 118343 , 71809 , 87161 , 139073 ) ( 188021 , 155527 , 86603 , 222791 ) ( 217823 , 191257 , 82913 , 261593 ) ( 57991 , 7739 , 75991 , 85921 ) ( 86467 , 40919 , 80251 , 107197 ) ( 102901 , 60713 , 82417 , 123271 ) ( 193357 , 171329 , 81241 , 233887 ) ( 50189 , 10141 , 68189 , 76319 ) ( 76169 , 41497 , 72833 , 96539 ) ( 175979 , 168187 , 73943 , 219749 ) ( 11191 , 7229 , 10081 , 14101 ) ( 93739 , 70493 , 72637 , 115909 ) ( 4787 , 1343 , 6787 , 7517 ) ( 823 , 691 , 599 , 1033 ) ( 15047 , 14771 , 7759 , 19217 ) ( 2669 , 523 , 4367 , 4679 ) ( 45011 , 19393 , 58937 , 67181 ) ( 68591 , 50269 , 64061 , 89321 ) ( 14141 , 13087 , 9623 , 18131 ) ( 117821 , 117283 , 67247 , 152591 ) ( 164921 , 180799 , 61331 , 219851 ) ( 27103 , 2183 , 52747 , 55033 ) ( 293 , 109 , 437 , 479 ) ( 46747 , 26627 , 57103 , 67477 ) ( 57901 , 41621 , 59749 , 78271 ) ( 70519 , 59039 , 62131 , 92689 ) ( 6541 , 6101 , 4909 , 8551 ) ( 121477 , 130157 , 63373 , 162007 ) ( 144271 , 161591 , 60379 , 195241 ) ( 22277 , 4201 , 46769 , 48407 ) ( 30767 , 14803 , 48767 , 52937 ) ( 5687 , 3859 , 7331 , 8597 ) ( 74567 , 77611 , 57959 , 102497 ) ( 108419 , 127303 , 57227 , 152189 ) ( 16123 , 449 , 43321 , 44053 ) ( 41161 , 33767 , 49963 , 61531 ) ( 51283 , 48857 , 52513 , 73453 ) ( 63013 , 66659 , 54511 , 89143 ) ( 134653 , 173627 , 48343 , 198223 ) ( 12497 , 2347 , 38183 , 38627 ) ( 19451 , 11101 , 39869 , 41621 ) ( 26669 , 21319 , 42251 , 47039 ) ( 34583 , 33433 , 44897 , 55313 ) ( 349 , 383 , 379 , 535 ) ( 54227 , 65077 , 49253 , 82157 ) ( 81839 , 109489 , 49481 , 125609 ) ( 99713 , 137563 , 46967 , 154643 ) ( 967 , 1361 , 337 , 1513 ) ( 20659 , 16271 , 38659 , 41389 ) ( 35599 , 39947 , 43783 , 57769 ) ( 44833 , 55421 , 45949 , 70963 ) ( 103609 , 151637 , 41893 , 167179 ) ( 1031 , 437 , 3733 , 3761 ) ( 3173 , 3659 , 4471 , 5543 ) ( 4063 , 5197 , 4733 , 6793 ) ( 1087 , 1661 , 509 , 1817 ) ( 11993 , 18527 , 4003 , 20123 ) ( 10631 , 8287 , 32243 , 32801 ) ( 16313 , 16657 , 34313 , 36683 ) ( 22403 , 26779 , 36791 , 43133 ) ( 47441 , 71977 , 42353 , 82211 ) ( 74093 , 118789 , 40181 , 129023 ) ( 11647 , 12569 , 31201 , 32377 ) ( 17041 , 21443 , 33487 , 37411 ) ( 30061 , 45599 , 38131 , 56191 ) ( 6991 , 11597 , 5713 , 12781 ) ( 61651 , 104333 , 38797 , 112621 ) ( 95791 , 163529 , 30001 , 174121 ) ( 1187 , 1861 , 3929 , 4097 ) ( 40919 , 79117 , 35213 , 84689 ) ( 52553 , 101911 , 33659 , 107483 ) ( 4747 , 9851 , 24919 , 25477 ) ( 8797 , 16973 , 26797 , 29167 ) ( 32437 , 68261 , 33109 , 72967 ) ( 4213 , 8741 , 2269 , 9103 ) ( 5531 , 16999 , 23531 , 26261 ) ( 13367 , 36259 , 27311 , 41297 ) ( 18761 , 49813 , 28517 , 53531 ) ( 60257 , 140269 , 18701 , 143987 ) ( 2521 , 13631 , 21139 , 22891 ) ( 5539 , 20849 , 22921 , 27709 ) ( 9013 , 30203 , 24727 , 35143 ) ( 107 , 337 , 209 , 365 ) ( 19057 , 57047 , 26683 , 59587 ) ( 26491 , 75401 , 25969 , 77461 ) ( 36109 , 97619 , 23551 , 99679 ) ( 48343 , 124133 , 18997 , 126673 ) ( 509 , 1243 , 95 , 1271 ) ( 281 , 1889 , 2281 , 2651 ) ( 109 , 437 , 293 , 479 ) ( 1487 , 5399 , 2731 , 5657 ) ( 5671 , 15343 , 1427 , 15601 ) ( 5027 , 29581 , 21389 , 32957 ) ( 8597 , 41143 , 22427 , 43367 ) ( 13439 , 55753 , 22577 , 57209 ) ( 39917 , 122191 , 13379 , 123647 ) ( 54167 , 153313 , 5657 , 155537 ) ( 1873 , 24629 , 19141 , 28003 ) ( 4309 , 15377 , 1933 , 15499 ) ( 641 , 34081 , 16889 , 35411 ) ( 3659 , 46699 , 16871 , 47429 ) ( 8093 , 62653 , 15677 , 63023 ) ( 115 , 659 , 103 , 661 ) ( 22937 , 106297 , 8033 , 106667 ) ( 34211 , 134851 , 719 , 135581 ) ( 8149 , 71567 , 12163 , 71719 ) ( 15103 , 93281 , 8089 , 93433 ) ( 2731 , 81059 , 2671 , 81061 ) ( 3457 , 515 , 3737 , 4541 ) ( 41017 , 22367 , 27557 , 46589 ) ( 15901 , 5567 , 18821 , 22145 ) ( 6943 , 6701 , 2831 , 8699 ) ( 8131 , 695 , 14957 , 15719 ) ( 425 , 379 , 319 , 547 ) ( 9643 , 6353 , 13331 , 15215 ) ( 19261 , 20507 , 15305 , 26849 ) ( 35143 , 43949 , 14039 , 50795 ) ( 4153 , 2153 , 10139 , 10397 ) ( 1525 , 1733 , 1751 , 2417 ) ( 3671 , 5311 , 1423 , 5869 ) ( 5353 , 11189 , 8495 , 12941 ) ( 1061 , 2263 , 1111 , 2419 ) ( 1861 , 3929 , 1187 , 4097 ) ( 5665 , 17399 , 6989 , 17957 ) ( 11 , 751 , 607 , 865 ) ( 121 , 2351 , 461 , 2357 ) ( 19391 , 4093 , 20269 , 25033 ) ( 12293 , 5431 , 14251 , 16999 ) ( 23423 , 19801 , 15793 , 29065 ) ( 13661 , 11521 , 12841 , 18367 ) ( 15077 , 18355 , 11311 , 22279 ) ( 29375 , 40213 , 9109 , 45001 ) ( 599 , 823 , 691 , 1033 ) ( 16973 , 26797 , 8797 , 29167 ) ( 3209 , 5587 , 6607 , 7915 ) ( 8579 , 16789 , 7573 , 17965 ) ( 19781 , 37711 , 4435 , 39463 ) ( 3329 , 9757 , 5869 , 10531 ) ( 9851 , 24919 , 4747 , 25477 ) ( 359 , 5269 , 4117 , 6001 ) ( 269 , 1195 , 319 , 1207 ) ( 8185 , 1013 , 7783 , 10069 ) ( 1241 , 355 , 983 , 1427 ) ( 86981 , 30685 , 63227 , 97949 ) ( 9413 , 5245 , 4859 , 10301 ) ( 2371 , 1415 , 1117 , 2599 ) ( 2435 , 1621 , 941 , 2693 ) ( 168605 , 117301 , 58403 , 187637 ) ( 22783 , 16475 , 7009 , 25531 ) ( 196871 , 147295 , 53177 , 222239 ) ( 61153 , 9299 , 58693 , 75577 ) ( 74491 , 23477 , 59491 , 86035 ) ( 89437 , 39503 , 60169 , 99829 ) ( 148243 , 102989 , 57163 , 165547 ) ( 203407 , 162473 , 46639 , 233959 ) ( 53087 , 11317 , 52067 , 66359 ) ( 65249 , 24655 , 53129 , 76217 ) ( 78971 , 39889 , 54023 , 89363 ) ( 112823 , 77773 , 53579 , 127247 ) ( 133817 , 101287 , 51377 , 152849 ) ( 158099 , 128425 , 47279 , 183467 ) ( 186101 , 159619 , 40853 , 219533 ) ( 607 , 11 , 751 , 865 ) ( 54967 , 18407 , 49585 , 66511 ) ( 81331 , 48227 , 51445 , 92299 ) ( 13939 , 9539 , 7333 , 15835 ) ( 13331 , 9643 , 6353 , 15215 ) ( 138337 , 113297 , 46855 , 161401 ) ( 18821 , 15901 , 5567 , 22145 ) ( 3979 , 205 , 5003 , 5731 ) ( 6311 , 2825 , 5263 , 7487 ) ( 11189 , 8495 , 5353 , 12941 ) ( 13375 , 11041 , 5111 , 15703 ) ( 1769 , 1555 , 517 , 2113 ) ( 4117 , 359 , 5269 , 6001 ) ( 37547 , 8023 , 42689 , 50819 ) ( 6607 , 3209 , 5587 , 7915 ) ( 58631 , 32467 , 45317 , 69023 ) ( 5501 , 3673 , 3551 , 6389 ) ( 86531 , 65647 , 46121 , 100955 ) ( 8009 , 6661 , 3443 , 9473 ) ( 14251 , 12293 , 5431 , 16999 ) ( 124703 , 111019 , 41645 , 150071 ) ( 20269 , 19391 , 4093 , 25033 ) ( 30217 , 3791 , 39433 , 44641 ) ( 39139 , 14033 , 40711 , 50683 ) ( 60511 , 39629 , 43339 , 71479 ) ( 107869 , 97283 , 40981 , 130933 ) ( 154681 , 153599 , 30169 , 194449 ) ( 25223 , 5329 , 34439 , 38495 ) ( 33161 , 14827 , 35885 , 44129 ) ( 42083 , 25933 , 37451 , 52475 ) ( 96251 , 95077 , 36275 , 121619 ) ( 116573 , 120703 , 31961 , 150005 ) ( 19429 , 1865 , 31567 , 33853 ) ( 2867 , 745 , 4019 , 4463 ) ( 34465 , 20261 , 34483 , 44857 ) ( 4853 , 3235 , 4409 , 6161 ) ( 43531 , 32015 , 35977 , 54499 ) ( 54205 , 46049 , 36919 , 67477 ) ( 66919 , 62795 , 36877 , 84223 ) ( 11345 , 11743 , 4253 , 14669 ) ( 121621 , 133625 , 26527 , 161389 ) ( 293 , 109 , 437 , 479 ) ( 21941 , 11029 , 28739 , 32909 ) ( 3163 , 1931 , 3697 , 4471 ) ( 36977 , 31441 , 31943 , 48521 ) ( 5197 , 4733 , 4063 , 6793 ) ( 46619 , 44923 , 32861 , 61043 ) ( 8119 , 8807 , 4021 , 10867 ) ( 72311 , 80599 , 31169 , 97679 ) ( 109427 , 130771 , 21893 , 152651 ) ( 1889 , 2281 , 281 , 2651 ) ( 11185 , 491 , 24877 , 25609 ) ( 16843 , 7373 , 26059 , 28387 ) ( 22957 , 15527 , 27697 , 33349 ) ( 29959 , 25385 , 29359 , 40927 ) ( 38281 , 37379 , 30613 , 51553 ) ( 3973 , 4763 , 1453 , 5581 ) ( 93409 , 115355 , 22909 , 133177 ) ( 114811 , 144509 , 15643 , 165523 ) ( 209 , 107 , 337 , 365 ) ( 379 , 319 , 425 , 547 ) ( 379 , 349 , 383 , 535 ) ( 383 , 379 , 349 , 535 ) ( 4409 , 4853 , 3235 , 6161 ) ( 5587 , 6607 , 3209 , 7915 ) ( 983 , 1241 , 355 , 1427 ) ( 8783 , 11243 , 2605 , 12839 ) ( 10897 , 14221 , 1931 , 16105 ) ( 8495 , 1741 , 21323 , 21767 ) ( 13361 , 7975 , 22577 , 24329 ) ( 18635 , 15529 , 24239 , 29027 ) ( 24749 , 24835 , 25877 , 36293 ) ( 7493 , 9655 , 3761 , 11117 ) ( 66245 , 88219 , 23549 , 99677 ) ( 83039 , 112765 , 18587 , 126263 ) ( 911 , 283 , 2465 , 2507 ) ( 9511 , 5087 , 20281 , 21055 ) ( 14137 , 11633 , 21799 , 24529 ) ( 2081 , 1933 , 2831 , 3389 ) ( 349 , 383 , 379 , 535 ) ( 25621 , 29837 , 24907 , 38893 ) ( 3659 , 4471 , 3173 , 5543 ) ( 33343 , 42359 , 25633 , 50647 ) ( 42961 , 57737 , 25231 , 66025 ) ( 54907 , 76403 , 23269 , 85459 ) ( 967 , 1361 , 337 , 1513 ) ( 69613 , 98789 , 19315 , 109381 ) ( 9767 , 13939 , 2057 , 15395 ) ( 87511 , 125327 , 12937 , 138223 ) ( 461 , 121 , 2351 , 2357 ) ( 6989 , 5665 , 17399 , 17957 ) ( 1111 , 1061 , 2263 , 2419 ) ( 15305 , 19261 , 20507 , 26849 ) ( 319 , 425 , 379 , 547 ) ( 27557 , 41017 , 22367 , 46589 ) ( 6743 , 10627 , 2789 , 11519 ) ( 60875 , 96511 , 15257 , 104099 ) ( 319 , 269 , 1195 , 1207 ) ( 7573 , 8579 , 16789 , 17965 ) ( 11311 , 15077 , 18355 , 22279 ) ( 15793 , 23423 , 19801 , 29065 ) ( 49801 , 83927 , 15745 , 89569 ) ( 64483 , 107993 , 10111 , 115195 ) ( 2393 , 4099 , 13205 , 13361 ) ( 5171 , 8677 , 14387 , 15563 ) ( 8213 , 14719 , 15833 , 19757 ) ( 11951 , 22657 , 17111 , 26375 ) ( 4523 , 8881 , 2231 , 9299 ) ( 42503 , 82009 , 11903 , 85727 ) ( 56201 , 105907 , 5861 , 110945 ) ( 319 , 2603 , 11821 , 11863 ) ( 509 , 1087 , 1661 , 1817 ) ( 5371 , 11399 , 13969 , 16339 ) ( 12247 , 27011 , 16213 , 29551 ) ( 33541 , 70769 , 12199 , 73309 ) ( 4883 , 10141 , 1247 , 10511 ) ( 95 , 509 , 1243 , 1271 ) ( 337 , 967 , 1361 , 1513 ) ( 941 , 2435 , 1621 , 2693 ) ( 1399 , 3541 , 1667 , 3727 ) ( 3973 , 8923 , 893 , 9181 ) ( 599 , 1285 , 19 , 1327 ) ( 19 , 599 , 1285 , 1327 ) ( 1103 , 6583 , 10721 , 11495 ) ( 517 , 1769 , 1555 , 2113 ) ( 5387 , 17587 , 12965 , 19811 ) ( 107 , 337 , 209 , 365 ) ( 8573 , 26149 , 13619 , 27605 ) ( 13031 , 37327 , 13385 , 38399 ) ( 19193 , 51553 , 11831 , 52625 ) ( 2881 , 7493 , 1303 , 7645 ) ( 509 , 1243 , 95 , 1271 ) ( 38357 , 90877 , 3035 , 93101 ) ( 1111 , 8705 , 10327 , 12079 ) ( 415 , 1997 , 1627 , 2311 ) ( 757 , 3035 , 1741 , 3229 ) ( 20401 , 59315 , 8677 , 60169 ) ( 599 , 13693 , 9371 , 15023 ) ( 5171 , 30169 , 9599 , 30539 ) ( 23729 , 77743 , 41 , 78473 ) ( 25 , 2393 , 1231 , 2497 ) ( 2113 , 24833 , 8599 , 25177 ) ( 437 , 3733 , 1031 , 3761 ) ( 5227 , 35627 , 7597 , 35779 ) ( 115 , 659 , 103 , 661 ) ( 9949 , 49565 , 5179 , 49717 ) ( 233 , 1033 , 59 , 1037 ) ( 16711 , 67079 , 913 , 67423 ) ( 1745 , 35149 , 4619 , 35177 ) ( 445 , 5207 , 421 , 5209 ) ( 5819 , 49015 , 1697 , 49043 ) ( 1753 , 41519 , 1705 , 41521 ) ( 54781 , 40727 , 13679 , 61667 ) ( 36259 , 27311 , 13367 , 41297 ) ( 11743 , 4253 , 11345 , 14669 ) ( 22657 , 17111 , 11951 , 26375 ) ( 40699 , 38393 , 8885 , 49961 ) ( 1873 , 1385 , 1409 , 2291 ) ( 14719 , 15833 , 8213 , 19757 ) ( 29377 , 35891 , 4787 , 41543 ) ( 2731 , 1487 , 5399 , 5657 ) ( 16999 , 23531 , 5531 , 26261 ) ( 3235 , 4409 , 4853 , 6161 ) ( 8677 , 14387 , 5171 , 15563 ) ( 3547 , 8027 , 4139 , 8585 ) ( 745 , 4019 , 2867 , 4463 ) ( 4099 , 13205 , 2393 , 13361 ) ( 529 , 7373 , 1889 , 7415 ) ( 2281 , 281 , 1889 , 2651 ) ( 24587 , 8587 , 15073 , 26647 ) ( 337 , 209 , 107 , 365 ) ( 6787 , 4787 , 1343 , 7517 ) ( 27011 , 16213 , 12247 , 29551 ) ( 46997 , 37279 , 8461 , 53857 ) ( 16193 , 8989 , 10351 , 18253 ) ( 30203 , 24727 , 9013 , 35143 ) ( 8807 , 4021 , 8119 , 10867 ) ( 3989 , 877 , 5983 , 6529 ) ( 20849 , 22921 , 5539 , 27709 ) ( 3673 , 4229 , 341 , 5003 ) ( 4733 , 4063 , 5197 , 6793 ) ( 11399 , 13969 , 5371 , 16339 ) ( 24761 , 32731 , 1009 , 36901 ) ( 1667 , 1399 , 3541 , 3727 ) ( 1087 , 1661 , 509 , 1817 ) ( 13631 , 21139 , 2521 , 22891 ) ( 1931 , 3697 , 3163 , 4471 ) ( 109 , 437 , 293 , 479 ) ( 2099 , 6883 , 2377 , 7039 ) ( 2603 , 11821 , 319 , 11863 ) ( 55385 , 11437 , 37937 , 60911 ) ( 82037 , 37225 , 36341 , 86699 ) ( 98783 , 53395 , 34211 , 104957 ) ( 141275 , 94447 , 25127 , 154361 ) ( 8167 , 2537 , 5041 , 8833 ) ( 69883 , 30257 , 34669 , 74329 ) ( 102295 , 62093 , 30481 , 110197 ) ( 146899 , 105929 , 19477 , 163441 ) ( 4229 , 341 , 3673 , 5003 ) ( 5311 , 1423 , 3671 , 5869 ) ( 8027 , 4139 , 3547 , 8585 ) ( 9757 , 5869 , 3329 , 10531 ) ( 437 , 293 , 109 , 479 ) ( 14201 , 10313 , 2341 , 15839 ) ( 17011 , 13123 , 1475 , 19297 ) ( 39509 , 8479 , 30743 , 45035 ) ( 3803 , 1429 , 2369 , 4145 ) ( 74843 , 44461 , 29105 , 81017 ) ( 110519 , 80809 , 22133 , 123605 ) ( 10253 , 7999 , 1199 , 11675 ) ( 32203 , 4727 , 28447 , 38377 ) ( 77731 , 51983 , 26119 , 85633 ) ( 94933 , 69857 , 22717 , 106507 ) ( 1241 , 355 , 983 , 1427 ) ( 2435 , 1621 , 941 , 2693 ) ( 4469 , 3775 , 443 , 5231 ) ( 26993 , 5977 , 24389 , 32519 ) ( 55127 , 36415 , 24407 , 61301 ) ( 68489 , 50929 , 22637 , 77471 ) ( 84563 , 68347 , 19259 , 97649 ) ( 103781 , 89101 , 13841 , 122267 ) ( 126575 , 113623 , 5951 , 151757 ) ( 21319 , 2897 , 22285 , 27493 ) ( 28153 , 10331 , 22843 , 32815 ) ( 36163 , 19181 , 23281 , 40609 ) ( 45781 , 29879 , 23167 , 51307 ) ( 57439 , 42857 , 22069 , 65341 ) ( 71569 , 58547 , 19555 , 83143 ) ( 88603 , 77381 , 15193 , 105145 ) ( 108973 , 99791 , 8551 , 131779 ) ( 607 , 11 , 751 , 865 ) ( 30521 , 19063 , 20159 , 35183 ) ( 39203 , 29257 , 20117 , 45377 ) ( 49877 , 41779 , 19043 , 58859 ) ( 98171 , 97633 , 5309 , 123353 ) ( 13123 , 1475 , 17011 , 19297 ) ( 40987 , 34715 , 18331 , 48889 ) ( 52333 , 48365 , 16561 , 63907 ) ( 83305 , 85097 , 7477 , 106111 ) ( 1555 , 517 , 1769 , 2113 ) ( 2831 , 2081 , 1933 , 3389 ) ( 3697 , 3163 , 1931 , 4471 ) ( 7783 , 8185 , 1013 , 10069 ) ( 10313 , 2341 , 14201 , 15839 ) ( 14771 , 7759 , 15047 , 19217 ) ( 2867 , 2071 , 2267 , 3533 ) ( 34913 , 33661 , 15713 , 43895 ) ( 58301 , 63289 , 10277 , 76787 ) ( 74279 , 83107 , 4499 , 99461 ) ( 7183 , 461 , 12625 , 13357 ) ( 11041 , 5111 , 13375 , 15703 ) ( 15499 , 10889 , 14293 , 19945 ) ( 20989 , 18227 , 14947 , 26515 ) ( 36793 , 39311 , 13735 , 48367 ) ( 61909 , 71819 , 6283 , 84715 ) ( 293 , 109 , 437 , 479 ) ( 1889 , 2281 , 281 , 2651 ) ( 8495 , 5353 , 11189 , 12941 ) ( 1751 , 1525 , 1733 , 2417 ) ( 41465 , 52363 , 8459 , 59951 ) ( 5869 , 3329 , 9757 , 10531 ) ( 691 , 599 , 823 , 1033 ) ( 12841 , 13661 , 11521 , 18367 ) ( 33199 , 44099 , 8947 , 49741 ) ( 44353 , 59957 , 4969 , 67159 ) ( 349 , 383 , 379 , 535 ) ( 967 , 1361 , 337 , 1513 ) ( 1889 , 529 , 7373 , 7415 ) ( 4139 , 3547 , 8027 , 8585 ) ( 1409 , 1873 , 1385 , 2291 ) ( 20099 , 30259 , 8963 , 33185 ) ( 2377 , 2099 , 6883 , 7039 ) ( 10351 , 16193 , 8989 , 18253 ) ( 15073 , 24587 , 8587 , 26647 ) ( 21547 , 35549 , 6913 , 38089 ) ( 103 , 115 , 659 , 661 ) ( 1271 , 2341 , 5561 , 5717 ) ( 2825 , 5263 , 6311 , 7487 ) ( 4787 , 9505 , 7037 , 10961 ) ( 7589 , 15499 , 7307 , 16571 ) ( 11663 , 23677 , 6689 , 24749 ) ( 17441 , 34471 , 4751 , 35927 ) ( 25355 , 48313 , 1061 , 50537 ) ( 1423 , 3671 , 5311 , 5869 ) ( 4939 , 11987 , 6499 , 12841 ) ( 12583 , 28271 , 4903 , 29125 ) ( 2719 , 5783 , 283 , 5977 ) ( 107 , 337 , 209 , 365 ) ( 509 , 1243 , 95 , 1271 ) ( 341 , 3673 , 4229 , 5003 ) ( 1343 , 6787 , 4787 , 7517 ) ( 205 , 5003 , 3979 , 5731 ) ( 1207 , 8765 , 4321 , 9109 ) ( 2953 , 14375 , 4111 , 14527 ) ( 10405 , 32867 , 307 , 33211 ) ( 115 , 659 , 103 , 661 ) ( 983 , 14041 , 2501 , 14069 ) ( 3401 , 21859 , 947 , 21887 ) ( 991 , 17531 , 955 , 17533 ) ( 13939 , 2057 , 9767 , 15395 ) ( 26149 , 13619 , 8573 , 27605 ) ( 46351 , 32741 , 3275 , 51263 ) ( 1361 , 337 , 967 , 1513 ) ( 15499 , 7307 , 7589 , 16571 ) ( 337 , 209 , 107 , 365 ) ( 29581 , 21389 , 5027 , 32957 ) ( 5003 , 3979 , 205 , 5731 ) ( 17587 , 12965 , 5387 , 19811 ) ( 4915 , 4409 , 23 , 5891 ) ( 3775 , 443 , 4469 , 5231 ) ( 9505 , 7037 , 4787 , 10961 ) ( 1769 , 1555 , 517 , 2113 ) ( 4471 , 3173 , 3659 , 5543 ) ( 1621 , 941 , 2435 , 2693 ) ( 383 , 379 , 349 , 535 ) ( 5263 , 6311 , 2825 , 7487 ) ( 1933 , 2831 , 2081 , 3389 ) ( 6583 , 10721 , 1103 , 11495 ) ( 355 , 983 , 1241 , 1427 ) ( 2341 , 5561 , 1271 , 5717 ) ( 599 , 1285 , 19 , 1327 ) ( 283 , 2465 , 911 , 2507 ) ( 3035 , 1741 , 757 , 3229 ) ( 11987 , 6499 , 4939 , 12841 ) ( 24629 , 19141 , 1873 , 28003 ) ( 1997 , 1627 , 415 , 2311 ) ( 425 , 379 , 319 , 547 ) ( 8705 , 10327 , 1111 , 12079 ) ( 3671 , 5311 , 1423 , 5869 ) ( 1061 , 2263 , 1111 , 2419 ) ( 11 , 751 , 607 , 865 ) ( 51833 , 16585 , 18419 , 53141 ) ( 64355 , 27187 , 15881 , 66239 ) ( 79445 , 39973 , 12023 , 82769 ) ( 42943 , 12401 , 17707 , 44251 ) ( 53857 , 21803 , 15793 , 55453 ) ( 67099 , 33197 , 12751 , 69847 ) ( 36581 , 13003 , 14801 , 37889 ) ( 46559 , 21901 , 13103 , 48443 ) ( 58817 , 32839 , 10229 , 62141 ) ( 10541 , 6607 , 821 , 11345 ) ( 38173 , 17285 , 12823 , 39769 ) ( 5399 , 2731 , 1487 , 5657 ) ( 48775 , 26927 , 10669 , 51523 ) ( 61849 , 38849 , 7099 , 66613 ) ( 77827 , 53483 , 1681 , 85471 ) ( 18527 , 4003 , 11993 , 20123 ) ( 3541 , 1667 , 1399 , 3727 ) ( 5983 , 3989 , 877 , 6529 ) ( 14221 , 1931 , 10897 , 16105 ) ( 1363 , 701 , 535 , 1447 ) ( 44581 , 31523 , 6073 , 49345 ) ( 8233 , 6323 , 241 , 9325 ) ( 1661 , 509 , 1087 , 1817 ) ( 2263 , 1111 , 1061 , 2419 ) ( 337 , 209 , 107 , 365 ) ( 437 , 293 , 109 , 479 ) ( 4019 , 2867 , 745 , 4463 ) ( 5269 , 4117 , 359 , 6001 ) ( 751 , 607 , 11 , 865 ) ( 11243 , 2605 , 8783 , 12839 ) ( 21479 , 13129 , 8339 , 23363 ) ( 28937 , 20803 , 7001 , 32261 ) ( 38531 , 30661 , 4343 , 44159 ) ( 8185 , 1013 , 7783 , 10069 ) ( 1241 , 355 , 983 , 1427 ) ( 2371 , 1415 , 1117 , 2599 ) ( 2435 , 1621 , 941 , 2693 ) ( 22783 , 16475 , 7009 , 25531 ) ( 4469 , 3775 , 443 , 5231 ) ( 3175 , 2771 , 121 , 3763 ) ( 607 , 11 , 751 , 865 ) ( 13331 , 9643 , 6353 , 15215 ) ( 18821 , 15901 , 5567 , 22145 ) ( 3737 , 3457 , 515 , 4541 ) ( 4117 , 359 , 5269 , 6001 ) ( 6607 , 3209 , 5587 , 7915 ) ( 14251 , 12293 , 5431 , 16999 ) ( 20269 , 19391 , 4093 , 25033 ) ( 2179 , 2213 , 103 , 2767 ) ( 2867 , 745 , 4019 , 4463 ) ( 4853 , 3235 , 4409 , 6161 ) ( 11345 , 11743 , 4253 , 14669 ) ( 293 , 109 , 437 , 479 ) ( 3163 , 1931 , 3697 , 4471 ) ( 5197 , 4733 , 4063 , 6793 ) ( 8119 , 8807 , 4021 , 10867 ) ( 1889 , 2281 , 281 , 2651 ) ( 209 , 107 , 337 , 365 ) ( 379 , 349 , 383 , 535 ) ( 983 , 1241 , 355 , 1427 ) ( 911 , 283 , 2465 , 2507 ) ( 2081 , 1933 , 2831 , 3389 ) ( 349 , 383 , 379 , 535 ) ( 3659 , 4471 , 3173 , 5543 ) ( 967 , 1361 , 337 , 1513 ) ( 9767 , 13939 , 2057 , 15395 ) ( 1111 , 1061 , 2263 , 2419 ) ( 319 , 425 , 379 , 547 ) ( 509 , 1087 , 1661 , 1817 ) ( 4883 , 10141 , 1247 , 10511 ) ( 19 , 599 , 1285 , 1327 ) ( 517 , 1769 , 1555 , 2113 ) ( 107 , 337 , 209 , 365 ) ( 2881 , 7493 , 1303 , 7645 ) ( 509 , 1243 , 95 , 1271 ) ( 437 , 3733 , 1031 , 3761 ) ( 115 , 659 , 103 , 661 ) ( 233 , 1033 , 59 , 1037 ) ( 445 , 5207 , 421 , 5209 ) ( 10141 , 1247 , 4883 , 10511 ) ( 13693 , 9371 , 599 , 15023 ) ( 2281 , 281 , 1889 , 2651 ) ( 6787 , 4787 , 1343 , 7517 ) ( 3673 , 4229 , 341 , 5003 ) ( 1087 , 1661 , 509 , 1817 ) ( 109 , 437 , 293 , 479 ) ( 1243 , 95 , 509 , 1271 ) ( 14375 , 4111 , 2953 , 14527 ) ( 7493 , 1303 , 2881 , 7645 ) ( 2393 , 1231 , 25 , 2497 ) ( 8765 , 4321 , 1207 , 9109 ) ( 1361 , 337 , 967 , 1513 ) ( 337 , 209 , 107 , 365 ) ( 5003 , 3979 , 205 , 5731 ) ( 1769 , 1555 , 517 , 2113 ) ( 383 , 379 , 349 , 535 ) ( 599 , 1285 , 19 , 1327 ) ( 30461 , 4105 , 7673 , 30647 ) ( 39335 , 10291 , 4871 , 39593 ) ( 50489 , 18085 , 893 , 51251 ) ( 24307 , 2081 , 7681 , 24565 ) ( 19913 , 3019 , 5807 , 20099 ) ( 35429 , 14311 , 851 , 36191 ) ( 15343 , 1427 , 5671 , 15601 ) ( 20929 , 5573 , 4189 , 21115 ) ( 28267 , 10991 , 1867 , 28813 ) ( 1285 , 19 , 599 , 1327 ) ( 2465 , 911 , 283 , 2507 ) ( 23729 , 10993 , 785 , 24491 ) ( 8923 , 893 , 3973 , 9181 ) ( 12877 , 4007 , 3091 , 13063 ) ( 18295 , 8249 , 1513 , 18841 ) ( 10139 , 4153 , 2153 , 10397 ) ( 14957 , 8131 , 695 , 15719 ) ( 5399 , 2731 , 1487 , 5657 ) ( 3541 , 1667 , 1399 , 3727 ) ( 5983 , 3989 , 877 , 6529 ) ( 437 , 293 , 109 , 479 ) ( 751 , 607 , 11 , 865 ) ( 1241 , 355 , 983 , 1427 ) ( 2435 , 1621 , 941 , 2693 ) ( 4469 , 3775 , 443 , 5231 ) ( 607 , 11 , 751 , 865 ) ( 293 , 109 , 437 , 479 ) ( 1889 , 2281 , 281 , 2651 ) ( 349 , 383 , 379 , 535 ) ( 967 , 1361 , 337 , 1513 ) ( 107 , 337 , 209 , 365 ) ( 509 , 1243 , 95 , 1271 ) ( 115 , 659 , 103 , 661 ) ( 1033 , 59 , 233 , 1037 ) ( 659 , 103 , 115 , 661 ) ( 3733 , 1031 , 437 , 3761 ) ( 1243 , 95 , 509 , 1271 ) ( 337 , 209 , 107 , 365 ) ( 5207 , 421 , 445 , 5209 ) ( 659 , 103 , 115 , 661 ) Have you reached the end of this post? I had to stop here. Otherwise Quora's mods could ban me because of spam. Don't forget to check every example individually, some of them recur:).
What are the two prime numbers whose difference is 10?
(3, 13) (7, 17) (13, 23) (19, 29) (31, 41) (37, 47) (43, 53) (61, 71) (73, 83) (79, 89) (97, 107) (103, 113) (127, 137) (139, 149) (157, 167) (163, 173) (181, 191) (223, 233) (229, 239) (241, 251) (271, 281) (283, 293) (307, 317) (337, 347) (349, 359) (373, 383) (379, 389) (409, 419) (421, 431) (433, 443) (439, 449) (457, 467) (499, 509) (547, 557) (577, 587) (607, 617) (631, 641) (643, 653) (673, 683) (691, 701) (709, 719) (733, 743) (751, 761) (787, 797) (811, 821) (829, 839) (853, 863) (877, 887) (919, 929) (937, 947) (967, 977) (1009, 1019) (1021, 1031) (1039, 1049) (1051, 1061) (1087, 1097) (1093, 1103) (1153, 1163) (1171, 1181) (1213, 1223) (1249, 1259) (1279, 1289) (1291, 1301) (1297, 1307) (1399, 1409) (1423, 1433) (1429, 1439) (1471, 1481) (1483, 1493) (1489, 1499) (1543, 1553) (1549, 1559) (1597, 1607) (1609, 1619) (1627, 1637) (1657, 1667) (1699, 1709) (1723, 1733) (1777, 1787) (1801, 1811) (1861, 1871) (1867, 1877) (1879, 1889) (1987, 1997) (1993, 2003) (2017, 2027) (2029, 2039) (2053, 2063) (2089, 2099) (2131, 2141) (2143, 2153) (2203, 2213) (2287, 2297) (2341, 2351) (2347, 2357) (2371, 2381) (2383, 2393) (2389, 2399) (2437, 2447) (2467, 2477) (2521, 2531) (2539, 2549) (2647, 2657) (2677, 2687) (2683, 2693) (2689, 2699) (2719, 2729) (2731, 2741) (2767, 2777) (2791, 2801) (2833, 2843) (2851, 2861) (2887, 2897) (2917, 2927) (2953, 2963) (3001, 3011) (3079, 3089) (3109, 3119) (3181, 3191) (3313, 3323) (3319, 3329) (3361, 3371) (3457, 3467) (3517, 3527) (3529, 3539) (3547, 3557) (3571, 3581) (3583, 3593) (3607, 3617) (3613, 3623) (3691, 3701) (3709, 3719) (3769, 3779) (3793, 3803) (3823, 3833) (3853, 3863) (3907, 3917) (3919, 3929) (4003, 4013) (4129, 4139) (4201, 4211) (4219, 4229) (4231, 4241) (4243, 4253) (4261, 4271) (4273, 4283) (4327, 4337) (4339, 4349) (4363, 4373) (4441, 4451) (4447, 4457) (4483, 4493) (4507, 4517) (4513, 4523) (4639, 4649) (4663, 4673) (4723, 4733) (4783, 4793) (4789, 4799) (4861, 4871) (4909, 4919) (4933, 4943) (4957, 4967) (4993, 5003) (4999, 5009) (5011, 5021) (5077, 5087) (5179, 5189) (5227, 5237) (5323, 5333) (5407, 5417) (5431, 5441) (5521, 5531) (5563, 5573) (5581, 5591) (5641, 5651) (5647, 5657) (5659, 5669) (5683, 5693) (5701, 5711) (5791, 5801) (5839, 5849) (5851, 5861) (5857, 5867) (5869, 5879) (6037, 6047) (6043, 6053) (6079, 6089) (6091, 6101) (6121, 6131) (6133, 6143) (6163, 6173) (6211, 6221) (6247, 6257) (6277, 6287) (6301, 6311) (6343, 6353) (6379, 6389) (6481, 6491) (6553, 6563) (6571, 6581) (6679, 6689) (6691, 6701) (6709, 6719) (6781, 6791) (6793, 6803) (6823, 6833) (6907, 6917) (6949, 6959) (6961, 6971) (6967, 6977) (6991, 7001) (7069, 7079) (7177, 7187) (7219, 7229) (7237, 7247) (7243, 7253) (7297, 7307) (7321, 7331) (7477, 7487) (7489, 7499) (7507, 7517) (7537, 7547) (7549, 7559) (7573, 7583) (7639, 7649) (7681, 7691) (7717, 7727) (7867, 7877) (7873, 7883)
What are the prime gaps of first 1000 prime numbers?
Third column shows the gaps:1, 2, 0 2, 3, 1 3, 5, 2 4, 7, 2 5, 11, 4 6, 13, 2 7, 17, 4 8, 19, 2 9, 23, 4 10, 29, 6 11, 31, 2 12, 37, 6 13, 41, 4 14, 43, 2 15, 47, 4 16, 53, 6 17, 59, 6 18, 61, 2 19, 67, 6 20, 71, 4 21, 73, 2 22, 79, 6 23, 83, 4 24, 89, 6 25, 97, 8 26, 101, 4 27, 103, 2 28, 107, 4 29, 109, 2 30, 113, 4 31, 127, 14 32, 131, 4 33, 137, 6 34, 139, 2 35, 149, 10 36, 151, 2 37, 157, 6 38, 163, 6 39, 167, 4 40, 173, 6 41, 179, 6 42, 181, 2 43, 191, 10 44, 193, 2 45, 197, 4 46, 199, 2 47, 211, 12 48, 223, 12 49, 227, 4 50, 229, 2 51, 233, 4 52, 239, 6 53, 241, 2 54, 251, 10 55, 257, 6 56, 263, 6 57, 269, 6 58, 271, 2 59, 277, 6 60, 281, 4 61, 283, 2 62, 293, 10 63, 307, 14 64, 311, 4 65, 313, 2 66, 317, 4 67, 331, 14 68, 337, 6 69, 347, 10 70, 349, 2 71, 353, 4 72, 359, 6 73, 367, 8 74, 373, 6 75, 379, 6 76, 383, 4 77, 389, 6 78, 397, 8 79, 401, 4 80, 409, 8 81, 419, 10 82, 421, 2 83, 431, 10 84, 433, 2 85, 439, 6 86, 443, 4 87, 449, 6 88, 457, 8 89, 461, 4 90, 463, 2 91, 467, 4 92, 479, 12 93, 487, 8 94, 491, 4 95, 499, 8 96, 503, 4 97, 509, 6 98, 521, 12 99, 523, 2 100, 541, 18 101, 547, 6 102, 557, 10 103, 563, 6 104, 569, 6 105, 571, 2 106, 577, 6 107, 587, 10 108, 593, 6 109, 599, 6 110, 601, 2 111, 607, 6 112, 613, 6 113, 617, 4 114, 619, 2 115, 631, 12 116, 641, 10 117, 643, 2 118, 647, 4 119, 653, 6 120, 659, 6 121, 661, 2 122, 673, 12 123, 677, 4 124, 683, 6 125, 691, 8 126, 701, 10 127, 709, 8 128, 719, 10 129, 727, 8 130, 733, 6 131, 739, 6 132, 743, 4 133, 751, 8 134, 757, 6 135, 761, 4 136, 769, 8 137, 773, 4 138, 787, 14 139, 797, 10 140, 809, 12 141, 811, 2 142, 821, 10 143, 823, 2 144, 827, 4 145, 829, 2 146, 839, 10 147, 853, 14 148, 857, 4 149, 859, 2 150, 863, 4 151, 877, 14 152, 881, 4 153, 883, 2 154, 887, 4 155, 907, 20 156, 911, 4 157, 919, 8 158, 929, 10 159, 937, 8 160, 941, 4 161, 947, 6 162, 953, 6 163, 967, 14 164, 971, 4 165, 977, 6 166, 983, 6 167, 991, 8 168, 997, 6 169, 1009, 12 170, 1013, 4 171, 1019, 6 172, 1021, 2 173, 1031, 10 174, 1033, 2 175, 1039, 6 176, 1049, 10 177, 1051, 2 178, 1061, 10 179, 1063, 2 180, 1069, 6 181, 1087, 18 182, 1091, 4 183, 1093, 2 184, 1097, 4 185, 1103, 6 186, 1109, 6 187, 1117, 8 188, 1123, 6 189, 1129, 6 190, 1151, 22 191, 1153, 2 192, 1163, 10 193, 1171, 8 194, 1181, 10 195, 1187, 6 196, 1193, 6 197, 1201, 8 198, 1213, 12 199, 1217, 4 200, 1223, 6 201, 1229, 6 202, 1231, 2 203, 1237, 6 204, 1249, 12 205, 1259, 10 206, 1277, 18 207, 1279, 2 208, 1283, 4 209, 1289, 6 210, 1291, 2 211, 1297, 6 212, 1301, 4 213, 1303, 2 214, 1307, 4 215, 1319, 12 216, 1321, 2 217, 1327, 6 218, 1361, 34 219, 1367, 6 220, 1373, 6 221, 1381, 8 222, 1399, 18 223, 1409, 10 224, 1423, 14 225, 1427, 4 226, 1429, 2 227, 1433, 4 228, 1439, 6 229, 1447, 8 230, 1451, 4 231, 1453, 2 232, 1459, 6 233, 1471, 12 234, 1481, 10 235, 1483, 2 236, 1487, 4 237, 1489, 2 238, 1493, 4 239, 1499, 6 240, 1511, 12 241, 1523, 12 242, 1531, 8 243, 1543, 12 244, 1549, 6 245, 1553, 4 246, 1559, 6 247, 1567, 8 248, 1571, 4 249, 1579, 8 250, 1583, 4 251, 1597, 14 252, 1601, 4 253, 1607, 6 254, 1609, 2 255, 1613, 4 256, 1619, 6 257, 1621, 2 258, 1627, 6 259, 1637, 10 260, 1657, 20 261, 1663, 6 262, 1667, 4 263, 1669, 2 264, 1693, 24 265, 1697, 4 266, 1699, 2 267, 1709, 10 268, 1721, 12 269, 1723, 2 270, 1733, 10 271, 1741, 8 272, 1747, 6 273, 1753, 6 274, 1759, 6 275, 1777, 18 276, 1783, 6 277, 1787, 4 278, 1789, 2 279, 1801, 12 280, 1811, 10 281, 1823, 12 282, 1831, 8 283, 1847, 16 284, 1861, 14 285, 1867, 6 286, 1871, 4 287, 1873, 2 288, 1877, 4 289, 1879, 2 290, 1889, 10 291, 1901, 12 292, 1907, 6 293, 1913, 6 294, 1931, 18 295, 1933, 2 296, 1949, 16 297, 1951, 2 298, 1973, 22 299, 1979, 6 300, 1987, 8 301, 1993, 6 302, 1997, 4 303, 1999, 2 304, 2003, 4 305, 2011, 8 306, 2017, 6 307, 2027, 10 308, 2029, 2 309, 2039, 10 310, 2053, 14 311, 2063, 10 312, 2069, 6 313, 2081, 12 314, 2083, 2 315, 2087, 4 316, 2089, 2 317, 2099, 10 318, 2111, 12 319, 2113, 2 320, 2129, 16 321, 2131, 2 322, 2137, 6 323, 2141, 4 324, 2143, 2 325, 2153, 10 326, 2161, 8 327, 2179, 18 328, 2203, 24 329, 2207, 4 330, 2213, 6 331, 2221, 8 332, 2237, 16 333, 2239, 2 334, 2243, 4 335, 2251, 8 336, 2267, 16 337, 2269, 2 338, 2273, 4 339, 2281, 8 340, 2287, 6 341, 2293, 6 342, 2297, 4 343, 2309, 12 344, 2311, 2 345, 2333, 22 346, 2339, 6 347, 2341, 2 348, 2347, 6 349, 2351, 4 350, 2357, 6 351, 2371, 14 352, 2377, 6 353, 2381, 4 354, 2383, 2 355, 2389, 6 356, 2393, 4 357, 2399, 6 358, 2411, 12 359, 2417, 6 360, 2423, 6 361, 2437, 14 362, 2441, 4 363, 2447, 6 364, 2459, 12 365, 2467, 8 366, 2473, 6 367, 2477, 4 368, 2503, 26 369, 2521, 18 370, 2531, 10 371, 2539, 8 372, 2543, 4 373, 2549, 6 374, 2551, 2 375, 2557, 6 376, 2579, 22 377, 2591, 12 378, 2593, 2 379, 2609, 16 380, 2617, 8 381, 2621, 4 382, 2633, 12 383, 2647, 14 384, 2657, 10 385, 2659, 2 386, 2663, 4 387, 2671, 8 388, 2677, 6 389, 2683, 6 390, 2687, 4 391, 2689, 2 392, 2693, 4 393, 2699, 6 394, 2707, 8 395, 2711, 4 396, 2713, 2 397, 2719, 6 398, 2729, 10 399, 2731, 2 400, 2741, 10 401, 2749, 8 402, 2753, 4 403, 2767, 14 404, 2777, 10 405, 2789, 12 406, 2791, 2 407, 2797, 6 408, 2801, 4 409, 2803, 2 410, 2819, 16 411, 2833, 14 412, 2837, 4 413, 2843, 6 414, 2851, 8 415, 2857, 6 416, 2861, 4 417, 2879, 18 418, 2887, 8 419, 2897, 10 420, 2903, 6 421, 2909, 6 422, 2917, 8 423, 2927, 10 424, 2939, 12 425, 2953, 14 426, 2957, 4 427, 2963, 6 428, 2969, 6 429, 2971, 2 430, 2999, 28 431, 3001, 2 432, 3011, 10 433, 3019, 8 434, 3023, 4 435, 3037, 14 436, 3041, 4 437, 3049, 8 438, 3061, 12 439, 3067, 6 440, 3079, 12 441, 3083, 4 442, 3089, 6 443, 3109, 20 444, 3119, 10 445, 3121, 2 446, 3137, 16 447, 3163, 26 448, 3167, 4 449, 3169, 2 450, 3181, 12 451, 3187, 6 452, 3191, 4 453, 3203, 12 454, 3209, 6 455, 3217, 8 456, 3221, 4 457, 3229, 8 458, 3251, 22 459, 3253, 2 460, 3257, 4 461, 3259, 2 462, 3271, 12 463, 3299, 28 464, 3301, 2 465, 3307, 6 466, 3313, 6 467, 3319, 6 468, 3323, 4 469, 3329, 6 470, 3331, 2 471, 3343, 12 472, 3347, 4 473, 3359, 12 474, 3361, 2 475, 3371, 10 476, 3373, 2 477, 3389, 16 478, 3391, 2 479, 3407, 16 480, 3413, 6 481, 3433, 20 482, 3449, 16 483, 3457, 8 484, 3461, 4 485, 3463, 2 486, 3467, 4 487, 3469, 2 488, 3491, 22 489, 3499, 8 490, 3511, 12 491, 3517, 6 492, 3527, 10 493, 3529, 2 494, 3533, 4 495, 3539, 6 496, 3541, 2 497, 3547, 6 498, 3557, 10 499, 3559, 2 500, 3571, 12 501, 3581, 10 502, 3583, 2 503, 3593, 10 504, 3607, 14 505, 3613, 6 506, 3617, 4 507, 3623, 6 508, 3631, 8 509, 3637, 6 510, 3643, 6 511, 3659, 16 512, 3671, 12 513, 3673, 2 514, 3677, 4 515, 3691, 14 516, 3697, 6 517, 3701, 4 518, 3709, 8 519, 3719, 10 520, 3727, 8 521, 3733, 6 522, 3739, 6 523, 3761, 22 524, 3767, 6 525, 3769, 2 526, 3779, 10 527, 3793, 14 528, 3797, 4 529, 3803, 6 530, 3821, 18 531, 3823, 2 532, 3833, 10 533, 3847, 14 534, 3851, 4 535, 3853, 2 536, 3863, 10 537, 3877, 14 538, 3881, 4 539, 3889, 8 540, 3907, 18 541, 3911, 4 542, 3917, 6 543, 3919, 2 544, 3923, 4 545, 3929, 6 546, 3931, 2 547, 3943, 12 548, 3947, 4 549, 3967, 20 550, 3989, 22 551, 4001, 12 552, 4003, 2 553, 4007, 4 554, 4013, 6 555, 4019, 6 556, 4021, 2 557, 4027, 6 558, 4049, 22 559, 4051, 2 560, 4057, 6 561, 4073, 16 562, 4079, 6 563, 4091, 12 564, 4093, 2 565, 4099, 6 566, 4111, 12 567, 4127, 16 568, 4129, 2 569, 4133, 4 570, 4139, 6 571, 4153, 14 572, 4157, 4 573, 4159, 2 574, 4177, 18 575, 4201, 24 576, 4211, 10 577, 4217, 6 578, 4219, 2 579, 4229, 10 580, 4231, 2 581, 4241, 10 582, 4243, 2 583, 4253, 10 584, 4259, 6 585, 4261, 2 586, 4271, 10 587, 4273, 2 588, 4283, 10 589, 4289, 6 590, 4297, 8 591, 4327, 30 592, 4337, 10 593, 4339, 2 594, 4349, 10 595, 4357, 8 596, 4363, 6 597, 4373, 10 598, 4391, 18 599, 4397, 6 600, 4409, 12 601, 4421, 12 602, 4423, 2 603, 4441, 18 604, 4447, 6 605, 4451, 4 606, 4457, 6 607, 4463, 6 608, 4481, 18 609, 4483, 2 610, 4493, 10 611, 4507, 14 612, 4513, 6 613, 4517, 4 614, 4519, 2 615, 4523, 4 616, 4547, 24 617, 4549, 2 618, 4561, 12 619, 4567, 6 620, 4583, 16 621, 4591, 8 622, 4597, 6 623, 4603, 6 624, 4621, 18 625, 4637, 16 626, 4639, 2 627, 4643, 4 628, 4649, 6 629, 4651, 2 630, 4657, 6 631, 4663, 6 632, 4673, 10 633, 4679, 6 634, 4691, 12 635, 4703, 12 636, 4721, 18 637, 4723, 2 638, 4729, 6 639, 4733, 4 640, 4751, 18 641, 4759, 8 642, 4783, 24 643, 4787, 4 644, 4789, 2 645, 4793, 4 646, 4799, 6 647, 4801, 2 648, 4813, 12 649, 4817, 4 650, 4831, 14 651, 4861, 30 652, 4871, 10 653, 4877, 6 654, 4889, 12 655, 4903, 14 656, 4909, 6 657, 4919, 10 658, 4931, 12 659, 4933, 2 660, 4937, 4 661, 4943, 6 662, 4951, 8 663, 4957, 6 664, 4967, 10 665, 4969, 2 666, 4973, 4 667, 4987, 14 668, 4993, 6 669, 4999, 6 670, 5003, 4 671, 5009, 6 672, 5011, 2 673, 5021, 10 674, 5023, 2 675, 5039, 16 676, 5051, 12 677, 5059, 8 678, 5077, 18 679, 5081, 4 680, 5087, 6 681, 5099, 12 682, 5101, 2 683, 5107, 6 684, 5113, 6 685, 5119, 6 686, 5147, 28 687, 5153, 6 688, 5167, 14 689, 5171, 4 690, 5179, 8 691, 5189, 10 692, 5197, 8 693, 5209, 12 694, 5227, 18 695, 5231, 4 696, 5233, 2 697, 5237, 4 698, 5261, 24 699, 5273, 12 700, 5279, 6 701, 5281, 2 702, 5297, 16 703, 5303, 6 704, 5309, 6 705, 5323, 14 706, 5333, 10 707, 5347, 14 708, 5351, 4 709, 5381, 30 710, 5387, 6 711, 5393, 6 712, 5399, 6 713, 5407, 8 714, 5413, 6 715, 5417, 4 716, 5419, 2 717, 5431, 12 718, 5437, 6 719, 5441, 4 720, 5443, 2 721, 5449, 6 722, 5471, 22 723, 5477, 6 724, 5479, 2 725, 5483, 4 726, 5501, 18 727, 5503, 2 728, 5507, 4 729, 5519, 12 730, 5521, 2 731, 5527, 6 732, 5531, 4 733, 5557, 26 734, 5563, 6 735, 5569, 6 736, 5573, 4 737, 5581, 8 738, 5591, 10 739, 5623, 32 740, 5639, 16 741, 5641, 2 742, 5647, 6 743, 5651, 4 744, 5653, 2 745, 5657, 4 746, 5659, 2 747, 5669, 10 748, 5683, 14 749, 5689, 6 750, 5693, 4 751, 5701, 8 752, 5711, 10 753, 5717, 6 754, 5737, 20 755, 5741, 4 756, 5743, 2 757, 5749, 6 758, 5779, 30 759, 5783, 4 760, 5791, 8 761, 5801, 10 762, 5807, 6 763, 5813, 6 764, 5821, 8 765, 5827, 6 766, 5839, 12 767, 5843, 4 768, 5849, 6 769, 5851, 2 770, 5857, 6 771, 5861, 4 772, 5867, 6 773, 5869, 2 774, 5879, 10 775, 5881, 2 776, 5897, 16 777, 5903, 6 778, 5923, 20 779, 5927, 4 780, 5939, 12 781, 5953, 14 782, 5981, 28 783, 5987, 6 784, 6007, 20 785, 6011, 4 786, 6029, 18 787, 6037, 8 788, 6043, 6 789, 6047, 4 790, 6053, 6 791, 6067, 14 792, 6073, 6 793, 6079, 6 794, 6089, 10 795, 6091, 2 796, 6101, 10 797, 6113, 12 798, 6121, 8 799, 6131, 10 800, 6133, 2 801, 6143, 10 802, 6151, 8 803, 6163, 12 804, 6173, 10 805, 6197, 24 806, 6199, 2 807, 6203, 4 808, 6211, 8 809, 6217, 6 810, 6221, 4 811, 6229, 8 812, 6247, 18 813, 6257, 10 814, 6263, 6 815, 6269, 6 816, 6271, 2 817, 6277, 6 818, 6287, 10 819, 6299, 12 820, 6301, 2 821, 6311, 10 822, 6317, 6 823, 6323, 6 824, 6329, 6 825, 6337, 8 826, 6343, 6 827, 6353, 10 828, 6359, 6 829, 6361, 2 830, 6367, 6 831, 6373, 6 832, 6379, 6 833, 6389, 10 834, 6397, 8 835, 6421, 24 836, 6427, 6 837, 6449, 22 838, 6451, 2 839, 6469, 18 840, 6473, 4 841, 6481, 8 842, 6491, 10 843, 6521, 30 844, 6529, 8 845, 6547, 18 846, 6551, 4 847, 6553, 2 848, 6563, 10 849, 6569, 6 850, 6571, 2 851, 6577, 6 852, 6581, 4 853, 6599, 18 854, 6607, 8 855, 6619, 12 856, 6637, 18 857, 6653, 16 858, 6659, 6 859, 6661, 2 860, 6673, 12 861, 6679, 6 862, 6689, 10 863, 6691, 2 864, 6701, 10 865, 6703, 2 866, 6709, 6 867, 6719, 10 868, 6733, 14 869, 6737, 4 870, 6761, 24 871, 6763, 2 872, 6779, 16 873, 6781, 2 874, 6791, 10 875, 6793, 2 876, 6803, 10 877, 6823, 20 878, 6827, 4 879, 6829, 2 880, 6833, 4 881, 6841, 8 882, 6857, 16 883, 6863, 6 884, 6869, 6 885, 6871, 2 886, 6883, 12 887, 6899, 16 888, 6907, 8 889, 6911, 4 890, 6917, 6 891, 6947, 30 892, 6949, 2 893, 6959, 10 894, 6961, 2 895, 6967, 6 896, 6971, 4 897, 6977, 6 898, 6983, 6 899, 6991, 8 900, 6997, 6 901, 7001, 4 902, 7013, 12 903, 7019, 6 904, 7027, 8 905, 7039, 12 906, 7043, 4 907, 7057, 14 908, 7069, 12 909, 7079, 10 910, 7103, 24 911, 7109, 6 912, 7121, 12 913, 7127, 6 914, 7129, 2 915, 7151, 22 916, 7159, 8 917, 7177, 18 918, 7187, 10 919, 7193, 6 920, 7207, 14 921, 7211, 4 922, 7213, 2 923, 7219, 6 924, 7229, 10 925, 7237, 8 926, 7243, 6 927, 7247, 4 928, 7253, 6 929, 7283, 30 930, 7297, 14 931, 7307, 10 932, 7309, 2 933, 7321, 12 934, 7331, 10 935, 7333, 2 936, 7349, 16 937, 7351, 2 938, 7369, 18 939, 7393, 24 940, 7411, 18 941, 7417, 6 942, 7433, 16 943, 7451, 18 944, 7457, 6 945, 7459, 2 946, 7477, 18 947, 7481, 4 948, 7487, 6 949, 7489, 2 950, 7499, 10 951, 7507, 8 952, 7517, 10 953, 7523, 6 954, 7529, 6 955, 7537, 8 956, 7541, 4 957, 7547, 6 958, 7549, 2 959, 7559, 10 960, 7561, 2 961, 7573, 12 962, 7577, 4 963, 7583, 6 964, 7589, 6 965, 7591, 2 966, 7603, 12 967, 7607, 4 968, 7621, 14 969, 7639, 18 970, 7643, 4 971, 7649, 6 972, 7669, 20 973, 7673, 4 974, 7681, 8 975, 7687, 6 976, 7691, 4 977, 7699, 8 978, 7703, 4 979, 7717, 14 980, 7723, 6 981, 7727, 4 982, 7741, 14 983, 7753, 12 984, 7757, 4 985, 7759, 2 986, 7789, 30 987, 7793, 4 988, 7817, 24 989, 7823, 6 990, 7829, 6 991, 7841, 12 992, 7853, 12 993, 7867, 14 994, 7873, 6 995, 7877, 4 996, 7879, 2 997, 7883, 4 998, 7901, 18 999, 7907, 6 1000, 7919, 12 Largest gap is 34, between 1327 and 1361.
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