Testwiki:Reference desk/Archives/Mathematics/2022 November 18

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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 $a{n}^2+bn+c$ 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, Template:Not a typo, or Template:Not a typo 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 $a{n}^{2k}+b{n}^k+c$ 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 Template:Not a typo 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)

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)

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)

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

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)

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)

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)