Form 6c: Fill & Download for Free

It’s probably the city of Sochi that was the site of the Winter Olympics in 2014. It’s the only place in Russia where palm trees can freely grow, due to Sochi’s warm climate. Right now, it’s -23C in Moscow and +6C in Sochi. It’s a popular Black Sea resort, but it also has a ski resort in the nearby mountains, hence the Winter Olympics.Pictured - The customary New Year palm tree in the city of Sochi.

To my CPU, this is a valid program:23 21 2f 75 73 72 2f 62 69 6e 2f 70 79 74 68 6f 6e 33 0a 70 72 69 6e 74 28 22 48 65 6c 6c 6f 22 29 0a It produces the following output:Hello To my CPU, this is also a valid program:23 21 2f 75 73 72 2f 62 69 6e 2f 70 79 74 68 6f 6e 33 0a 70 72 69 6e 74 20 22 48 65 6c 6c 6f 22 20 0a It produces the following output: File "./prog", line 2  print "Hello"  ^ SyntaxError: Missing parentheses in call to 'print' Ugh, okay. So “all possible programs” seems not that helpful. Hold that thought for a moment, will you?“Wait a second!”, little Timmy says, “That is __obviously__ not valid! The CPU should just reject it!”Let’s say we teach our CPU to check a program. Let’s also assume that it can actually understand the above output means “not valid”. With no time overhead.The CPU runs a program and just “knows” it is valid. Simple enough.To my CPU, validating either program takes this long:0m0.053s Ugh, okay. So “20 programs per second” seems not that helpful. Let’s say we have a compute farm 100.000.000 times as large. Or a 4GHz 544 bit Halting Machine[math]^\text{TM}[/math] that just needs one cycle to load, and one cycle to check our program.Then we could validate 2.000.000.000 programs per second. Hold that thought for a moment, will you?“Wait a second!”, little Timmy says, “That is __obviously__ too generous! What are you up to?”Let’s say we only care about 34 character long programs. Let’s say they must be printable. How many programs could 34 characters from a 95-sized alphabet[1][1][1][1] form?I asked my CPU. It said this:17482461472379729163335779494664534419607208692468702793121337890625 Ugh, okay. Just hold that number for a second. We are almost done.“Wait a second!”, little Timmy begs, “Can we just, say, stop, and not…”How long does our magic Halting Machine[math]^\text{TM}[/math] or magic multi-million Dollar compute farm need for this?I asked my CPU. It said this:8741230736189865032833812947758855352681500222502396231680 seconds 2428119648941629319461947407662588838058584310775021568 hours 101171652039234547888365164466389879432136691453788160 days 277182608326669973730021063496542724241704047607808 years 19424957191533030140568429216763039711232 ages of the universe Better take a human for this. They perform better.[2][2][2][2]Footnotes[1] ASCII - Wikipedia[1] ASCII - Wikipedia[1] ASCII - Wikipedia[1] ASCII - Wikipedia[2] "Hello, World!" program - Wikipedia[2] "Hello, World!" program - Wikipedia[2] "Hello, World!" program - Wikipedia[2] "Hello, World!" program - Wikipedia

Plainly, [math](0,0,0)[/math] is a solution to this equation in integers. It is easy to verify that there is no other solution where one of [math]a,b,c = 0[/math].Hereafter, we assume without loss of generality that [math]a,b,c > 0[/math]. We claim that there is no such solution.Suppose to the contrary that [math](a,b,c)[/math] is the solution in positive integers such that [math]2a^2 [/math][math][/math][math]+ 3b^2 = 6c^2[/math] with [math]c[/math] as small as possible. Since [math]3 \mid 3b^2[/math] and [math]3\mid 6c^2[/math], we must have [math]3 \mid 2a^2[/math] and thus [math]3 \mid a[/math].Writing [math]a = 3A[/math] for some positive integer [math]A[/math], we obtain[math]6A^2 [/math][math][/math][math]+ b^2 = 2c^2. \tag*{}[/math]Reducing this equation modulo [math]3[/math], this yields[math]b^2 \equiv 2c^2 \bmod 3, \tag*{}[/math]which is only true when [math]b \equiv c \equiv 0 \bmod 3[/math].Writing [math]b = 3B[/math] and [math]c = 3C[/math] for some positive integers [math]B, C[/math], we obtain[math]2A^2 [/math][math][/math][math]+ 3B^2 = 6C^2. \tag*{}[/math]Hence, [math](A,B,C)[/math] is another solution to our diophantine equation in question where [math]C < c[/math]. This contradicts the minimality of [math]c[/math], and thus there are no solutions to the given diophantine equation where [math]a,b,c \neq 0.[/math]Therefore, the diophantine equation [math]2a^2 [/math][math][/math][math]+ 3b^2 = 6c^2[/math] has only one solution in integers [math](a,b,c) = (0,0,0)[/math].