Question

Of all positive integers x, for which x is x^2+1 a prime number? (Hint: use induction, starting with 1^2+1=2 is prime.)

Answer 1

I think I can prove that all of the rest must be even

Answer 2

not all evens satisfy the condition, though.

Answer 3

\[(x + n)^2 + 1 = P => x^2 + 2xn + n^2 + 1 = P \]

Answer 4

\[\frac{P-1}{2} = \frac{x^2 + n^2}{2} + xn\]

Answer 5

\[\frac{P-1}{2} \]
is an integer
\[xn\]
is an integer
\[=> \frac{n^2 +x^2}{2} \]
is an integer

Answer 6

\[=> x^2 + n^2 \]
must be even

Answer 7

=> if x is odd, n is odd and if x is even, n is even

Answer 8

but once you increment by an odd, you end up at an even

Answer 9

and when you are at an even, incrementing by an even gives you an even still

Answer 10

of course, I left out x = 1

Answer 11

1 is still odd :P