Question

Show that 13 is the only prime that divides two successive integers of the form n
^2 + 3
​

Answer 1

Answer:

Let an=n2+3. If p|an then n2=−3(modp).

A quick calculation shows that an+1−an=2n+1 so if p also divides an+1 we must have 2n=−1(modp). Square this to see that it implies 4n2=1⟹−12=1(modp). Thus, p|13 and we are done.

Answer 2

Answer:

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