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 does awfully well: it produces primes for :

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.)

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Analytic Class Number Formula | Mental Wildernesson June 27, 2014at 2:51 pm

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

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JoseBroxon September 23, 2017at 7:00 pm

@JoseBrox Updated!

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holdenleeon September 24, 2017at 4:09 am