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I was doing a problem in my Discrete Mathematics book and it called for finding an infinite number of counterexamples to the statement "7n+2" is a perfect square (which fails for n=3 at least).

In my search for such an infinite counterexample, I tried to find A, n=n(k,A), such that 7n+2=49k^{2}-A^{2}for some A, and for all k. Then, I would be able to say that for n of the form, n=(k,A), where A is some fixed number, 7n+2=49k^{2}-A^{2}, which is a perfect square (7k)^{2}minus the fixed number A^{2}. Clearly, if you can find this then you're done because squares will get large enough that the distance between them is more than A^{2}, and thus, for a large enough n, 7n+2=49k^{2}-A^{2}would not be a perfect square, as the next lowest square would be more than A^{2}away.

Anyway, in my search for this number, I let n be a quadratic function of k, n = xk^{2}+yk+z for some fixed integers x, y, z to be determined, and from there I got these equations:

7(xk^{2}+yk+z)+2=49k^{2}-A^{2}, which gives

7x=49 => x=7

yk=0 => y=0

7z+2=-A^2 => z=(-A^{2}-2)/7

Now the first two equations give quick answers for x and y, but in order for 7 to be an integer, A^{2}+2 must be a multiple of 7.

This is where I was led to the current title of this thread.

Now, I've put the first 1000 numbers of the form A^{2}+2 (i.e. A=1, 2, 3, ..., 1000) into Excel and not a single one of them has been divisible by 7. Since, statistically, every 7th number should be divisible by 7, if we consider A^{2}+2 to be a fairly random number, I thought that there must be something odd going on here that none of them had been a multiple of 7.

I could continue to put numbers into Excel (checking the first 2000, 5000, 10 000, and so on), but I thought I'd ask people here if they thought there was a simple number theory explanation for this. It's particularly pertinent for me since my whole proof hinges on an integer z existing. And if it doesn't, I shall have to try another route.

Thanks,

Chris

P. S. Also, if this statement is true (my conjecture), then it is either something that has already been proved or a pretty interesting result (to me at least), so I'd appreciate it if someone was able to prove the conjecture or point out where it's already been proven.

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# Conjecture: There exists no number k s.t. k^2+2 is a multiple of 7

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