Prime rich and prime poor

A short excursion –

The well-known Euler’s Polynomial $latex x^2 – x + 41$ 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 $latex (x^5 – 133 c^4 + 6729 x^3 – 158379 x^2 + 1720294x – 6823316)/4$, 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 $latex x^{12} + 488669$ doesn’t produce a prime until $latex x = 616980$. 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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