Posted by: holdenlee | March 16, 2014

## Prime-producing polynomials

No nonconstant polynomial in one variable can produce only primes at integer values. But the polynomial $x^2+x+41$ does awfully well: it produces primes for $0\le x\le 39$:

41 ,

43 ,

47 ,

53 ,

61 ,

71 ,

83 ,

97 ,

113 ,

131 ,

151 ,

173 ,

197 ,

223 ,

251 ,

281 ,

313 ,

347 ,

383 ,

421 ,

461 ,

503 ,

547 ,

593 ,

641 ,

691 ,

743 ,

797 ,

853 ,

911 ,

971 ,

1033 ,

1097 ,

1163 ,

1231 ,

1301 ,

1373 ,

1447 ,

1523.

Why is this the case? We’ll explore these remarkable prime-producing polynomials here (source, template). Note that you will need to know enough abstract algebra/algebraic number theory to understand factorization in quadratic rings.

(I found out about these polynomials from the post here.)

## Responses

1. […] relates a class number to the distribution of quadratic residues modulo a prime p. In the post on prime-producing polynomials I noted that when the class number is 1, more quadratic nonresidues clump near the beginning. The […]

2. Unfortunately, the links are not working anymore. Can you fix them? Thanks!

3. @JoseBrox Updated!