Are numbers of the form $n^2+n+17$ always prime Someone claimed that a number, multiplied by the number after it plus 17 is always prime, and showed several cases. I'm not a complete amateur in Number Theory, and I know that $17*18+17=17*19$, so it does not work for $n\equiv0(\mod17)$ but does it always work for other $n$ values? If not, can someone give me a counter example that I can show that person so they can correct their statement?

There are plenty of numbers besides multiples of $17$ that fail to give primes in that formula.
Even with "handicaps" like the one you give, there's just no polynomial that always gives primes. according to Mathworld, Legendre proved this long, long ago: http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
See Sloane's A007636.
