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November 18

Requirements a polynomial must have to produce infinitely many primes

Linear polynomials $an+b$ merely need to have a and b relatively prime.

Quadratic polynomials $an^{2}+bn+c$ require a, b, and c to have no common factors greater than 1, but also for $b^{2}-4ac$ to NOT be a perfect square.

Any requirements a cubic, quartic, quintic, sextic, septimic, or octavic polynomial must have?? (What I know is that the GCF of all coefficients must be 1 in a polynomial of any degree, and also that for a polynomial of the form $an^{2k}+bn^{k}+c$ there's the requirement that $b^{2}-4ac$ cannot be a perfect square. Any other requirements?? (Also there's the requirement that for cubic, quintic, and septimic polynomials to produce infinitely many primes, it cannot be factored by grouping it as the product of a linear binomial and a polynomial with all even exponents.) Georgia guy (talk) 02:29, 18 November 2022 (UTC)[reply]

The stated conditions are necessary but not sufficient for polynomials of higher degrees. The polynomial

n^{3}-n+3

fulfills the conditions but produces no other primes than

3.

The precise criterion is open; see Bunyakovsky conjecture. Note that

n^{3}-n+3

fails Bunyakovsky's third condition. --Lambiam 03:43, 18 November 2022 (UTC)[reply]

It's still an open problem for n²+1. The fact about linear polynomials is called Dirichlet's theorem on arithmetic progressions and it's kind of a big deal, a conjecture for a long time, special cases proven, then finally the full theorem with help from an unexpected direction. Basically it was the nucleus of a new branch of mathematics, analytic number theory; new branches of mathematics aren't created every day. The original proof is available on-line btw, and it's probably not inaccessible to ordinary mortals provided they are familiar with complex variables, Euler's proof of the infinitude of primes, and, of course, German. --RDBury (talk) 05:08, 18 November 2022 (UTC)[reply]

Usually, degrees 7 and 8 are called septic and octic, respectively. Double sharp (talk) 01:57, 20 November 2022 (UTC)[reply]

Is it for any natural reason or simply because 2-syllable words are easier to predict?? Georgia guy (talk) 02:08, 20 November 2022 (UTC)[reply]

I'm pretty sure these terms date from medieval Latin, but you can find etymologies in Wiktionary. --RDBury (talk) 03:35, 20 November 2022 (UTC)[reply]

Wiktionary treats them as if formed in English. *Septicus and *octicus are not found in Du Cange's Glossarium mediae et infimae Latinitatis. --Lambiam 04:49, 20 November 2022 (UTC)[reply]