A question on the prime generating polynomial $x^2 -79x+1601$. In Tom Apostol's book Analytic Number Theory, author says $x^2 -79x+1601$ gives prime numbers for $x=0,1,...,79$.
We can see this by putting values. Is there any other way of knowing this property of this expression? How can we construct more expression with this property? 
 A: There is a good deal of repetition involved here. If we take the pure translation
$$ x = t + 40 $$ the polynomial becomes
$$ t^2 + t + 41 $$
which is prime for $0 \leq t \leq 39.$ It is also prime for $-40 \leq t \leq -1,$ however these primes are exactly the same primes as thos for non-negative $t.$
The reason we know that this polynomial ($t$) gives all those prime values is that $41$ is prime, $4 \cdot 41 - 1 = 163$ is also prime, and the positive binary quadratic form $$ x^2 + xy + 41 y^2$$ is of class number one, meaning all forms of that discriminant are ($SL_2 \mathbb Z$) equivalent to it. The basic facts here were known to Gauss, an elementary proof of if and only if was provided by Rabinowitz in 1913. See my proof at Is the notorious $n^2 + n + 41$ prime generator the last of its type?
For those who do not know anything about binary quadratic forms, this is a table from Rose, A Course in Number Theory The ordered triple $A,B,C$ refers to the (positive) quadratic form
$$ f(x,y) = A x^2 + B xy + C y^2. $$ See how there is just one entry for discriminant $-163.$

