prime numbers

[pale cave]

Find all prime numbers p and q such that $5p^2 - 2q^2 = 6q - 9p$
and
Is there a number n such that the numbers
n-1996, n, n+1996
are prime? (All prime numbers are positive.)
V. Senderov

[royal marsh]

tippy

[wicked nova]

2B.

So we have these three numbers, which actually form an arithmetic sequence of three terms with difference 1996.

1996 % 3 = 1, & we can see that no matter what, at one of the three numbers has to be divisible by 3. As a positive prime number, this number must be 3.

Since all of these numbers have to be positive & n - 1996 is the smallest number, n - 1996 = 3, & n = 1999. However, 1999 + 1996 = 3995, which is clearly divisible by 5.

Thus, there is a contradiction. There is no such number n.

Unfortunately wasn't quite sure how to explain it without just giving you the answer.

[pale cave]

thanks!

[pale cave]

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