Explorations in math and programming
David Lowry-Duda

A short excursion -

The well-known Euler's Polynomial $x^2 - x + 41$ generates 40 primes at the first 40 natural numbers. It is sometimes called a prime-rich polynomial. There are many such polynomials, and although Euler's Polynomial is perhaps the best-known, it is not the best. The best that I have heard of is $(x^5 - 133 c^4 + 6729 x^3 - 158379 x^2 + 1720294x - 6823316)/4$, which generates 57 primes. But this morning, I was reading an article on Ulam's Spiral when I heard of the opposite - a prime-poor polynomial. The polynomial $x^{12} + 488669$ doesn't produce a prime until $x = 616980$. Who knew?

And to give them credit, that prime-rich polynomial was first discovered by Jaroslaw Wroblewski & Jean-Charles Meyrignac in one of Al Zimmerman's Programming Contests (before being found by a few other teams too).

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Comments (1)
  1. 2021-12-25 Vaskor Basak

    What polynomials are allowable for prime-poor polynomials? Could I claim that I have found a better example of a prime-poor polynomial than $x^{12}+488669$ by presenting the example $(x+3)^{12}-488601$, for example?