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Prove that the equation $x^2 − y^2 = 2002$ has no integer solution.

If I had the equation $x^3 - 7y = 3$ I would easily conclude that for it to have any integer solution then $$x^2\equiv3\pmod 7$$ must be true.

Applying the same logic here, I know that $x² = y^2 +2002$ so I can conclude that for it to have any integer solution then $$x^2\equiv2002\pmod y$$ must be true.

Is this conclusion correct?

From that congruence I can prove that the equation hasn't any integer solution without much trouble but I am not sure if that congruence is valid.

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    $\begingroup$ $x^2\equiv2002\pmod y$ follows from the equation, but it doesn't appear to get you closer to a solution. $\endgroup$ – Joffan Mar 17 '17 at 17:51
  • $\begingroup$ @Joffan for any x $$x\equiv0,1,...,y-1\pmod y$$ so $$x²\equiv2002\pmod y$$ is equivalent to $$(y-1)²\equiv2002\pmod y$$ which must be true if there are any integer solutions. $$(y-1)²\equiv y²+1-2y\pmod y\equiv 1\pmod y$$ 1 is different than 2002 so the equation doesn't have any integer solution. Am I thinking correctly? $\endgroup$ – Cat_astrophic Mar 17 '17 at 18:17
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    $\begingroup$ No your inference of $(y-1)²\equiv2002\pmod y$ doesn't follow from the previous. $\endgroup$ – Joffan Mar 17 '17 at 18:56
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    $\begingroup$ Well, you can do that, but it doesn't eliminate the possibility that there are other values of $y$ that will work. $\endgroup$ – Joffan Mar 17 '17 at 19:25
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    $\begingroup$ @Joffan Oh, yes you are absolutely right. Thank you very much. $\endgroup$ – Cat_astrophic Mar 17 '17 at 19:32
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One way to solve this is to look at $x^2-y^2=(x+y)(x-y)$. Then for integer $x,y$, since $2002$ is even, one of $(x+y),(x-y)$ must be even, but since $2002/2=1001$, the other must be odd. That would mean $(x+y)+(x-y)=2x$ is also odd, which contradicts the existence of integer solutions.

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    $\begingroup$ In fact if one of $x+y$, $x-y$ is even, then both of them are, but then their product would be a multiple of $4$, which $2002$ is not. $\endgroup$ – Jack M Mar 22 '17 at 20:40
  • $\begingroup$ You can take it either way around. I went for the contradiction. $\endgroup$ – Joffan Mar 22 '17 at 20:40
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Hint:

$a\equiv0,1,2,3\pmod4, a^2\equiv0,1$

Or

$x+y+(x-y)$ is even, $x+y, x-y$ have the same parity

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  • $\begingroup$ I know why the mod 4 works but what about mod y? $$x\equiv0, 1, ..., (y-1)\pmod y$$ $\endgroup$ – Cat_astrophic Mar 17 '17 at 17:31
  • $\begingroup$ @Cat_astrophic I think lab bhattacharjee's point is that, although your result in the question is true, it's still going to be a lot of work for you to attempt to prove that no solutions exist. If you consider things mod 4, there are a small integer number of cases to disprove (i.e you can actually do it in two). $\endgroup$ – origimbo Mar 17 '17 at 17:52
  • $\begingroup$ @tatan, if the sum of two numbers is even, either both are odd, or both even $\endgroup$ – lab bhattacharjee Nov 1 '18 at 14:47
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Answering this requires no advanced math knowledge at all.

1     4     9     16     25  
   3     5     7      9

The difference between every square is an odd number, and since 2002 is even, then y must be an even number less than x. Therefore for there to be an integer solution we have to express 2002 as the sum of an even number of consecutive odd numbers.

3 + 5                   = 8            
3 + 5 + 7 + 9           = 24
    5 + 7               = 12     
    5 + 7 + 9 + 11      = 32
        7 + 9           = 16           
        7 + 9 + 11 + 13 = 40 

These sequences.of sums can be Written as 4n+4 and 8n+16 respectively which we can then write as 4 (n+1) and 4(2n+4)

And so on. The sequence of the sum of 6 and 8 and every other number of consecutive odd numbers will also be able to be Written as 4(kn+k^2) where k is half the number of consecutive odd numbers you are adding up. 2002 is not divisible by 4, So either k or n must not be an integer number, which means that there is no integer solutions.

Therefore

All even numbers not divisible by 4 cannot be Written as the difference between two squares

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Given that n, a, and b are elements of N, let n = a*b. (If n is prime, then n may be written as n*1 = 1*n; if n is non-prime, then by definition n has at least two divisors.)

n = ab

n = 0 + 4ab/4 + 0

0 = (a^2-a^2)/4 and 0 = (b^2-b^2)/4

n = (a^2-a^2)/4 +(2ab+2ab)/4 + (b^2-b^2)/4

n = (a^2 - a^2 + 2ab + 2ab + b^2 + b^2 - b^2)/4

rearranging and regrouping, we get n = (a^2 + 2ab + b^2)/4 - (a^2 - 2ab + b^2)/4

simplifying, we get n = ((a+b)/2)^2 - ((a-b)/2)^2

n = c^2 - d^2

n may be written as the difference of two numbers c and d. Both c and d are members of the natural numbers when a and b are odd or when both a and b are even. When n is divisible by 2 but not by 4. c^2 and d^2 must have the form (4m+2)/4 and thus cannot be members of N.

In this specific case, the divisors of 2002 are 2, 7, 11, and 13. Any possible solutions of a and b must be made with a combination of these primes. Since 2002 is divisible by 2 and not by 4, the solutions of a and b must be of the form SQRT[(4m+2)/4] which equals SQRT[m + .5], which, sadly, is not an element of N.

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