This is the original question. Aside from simply calculating the values of $p(0)$ through $p(40)$, can anyone suggest how modular arithmetic might be used to show that $p(x)$ returns primes for all values of $x$ in the set $\{0, 1, 2, ..., 40\}$? For context, this question comes from a relatively introductory math text, though it is marked as a challenging question.
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If we don't want to use brute force, we can do this via some simple modular arithmetic plus the magical fact that $\mathbb Z[\frac{1 + \sqrt{-163}}{2}]$ is a unique factorization domain. Proving this fact is hard, though. – Misha Lavrov Nov 25 '19 at 01:30
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1I give an elementary proof of an if and only if version at https://math.stackexchange.com/questions/289338/is-the-notorious-n2-n-41-prime-generator-the-last-of-its-type/289357#289357 The original was by Rabinowitz in 1913. – Will Jagy Nov 25 '19 at 01:43
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@MishaLavrov the whole business is fairly easy when written using binary quadratic forms. Of course, that is of no help for https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner_theorem – Will Jagy Nov 25 '19 at 02:36
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I prefer to write $x^2 + x + 41.$ Just habit. Same values, note $$ (x+1)^2 - (x+1) + p = x^2 + x + p $$
The statement is this: given a (positive) prime number $p,$ the unary polynomial $$ x^2 + x + p $$ takes prime values for all integer $x$ with $0 \leq x \leq p-2$ if, and only if, the (positive) binary quadratic form $$ x^2 + xy + p y^2 $$ is the only form (up to $SL_2 \mathbb Z$ equivalence) of its discriminant. Checking class number of a discriminant $\Delta$ is a finite computation, we need only consider Gauss reduced forms, meaning $$ a x^2 + b xy + c y^2 $$ with $$ b^2 - 4ac = \Delta \; , \; $$ $$ \gcd(a,b,c) = 1 \; , \; $$ $$ |b| \leq a \leq c $$ IF $a=c,$ then $$ b \geq 0 $$ IF $|b| = a,$ then $$ b = a $$
Will Jagy
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