Editing Ulam spiral (section)

==Explanation==
Diagonal, horizontal, and vertical lines in the number spiral correspond to polynomials of the form
: <math>f(n) = 4 n^2 + b n + c</math>
where ''b'' and ''c'' are integer constants.  When ''b'' is even, the lines are diagonal, and either all numbers are odd, or all are even, depending on the value of ''c''.  It is therefore no surprise that all primes other than 2 lie in alternate diagonals of the Ulam spiral.  Some  polynomials, such as <math>4 n^2 + 8 n + 3</math>, while producing only odd values, factorize over the integers <math>(4 n^2 + 8 n + 3)=(2n+1)(2n+3)</math> and are therefore never prime except possibly when one of the factors equals 1. Such examples correspond to diagonals that are devoid of primes or nearly so.

To gain insight into why some of the remaining odd diagonals may have a higher concentration of primes than others, consider <math>4 n^2 + 6 n + 1</math> and <math>4 n^2 + 6 n + 5</math>. Compute remainders upon division by 3 as ''n'' takes successive values 0, 1, 2, ....  For the first of these polynomials, the sequence of remainders is 1, 2, 2, 1, 2, 2, ..., while for the second, it is 2, 0, 0, 2, 0, 0, .... This implies that in the sequence of values taken by the second polynomial, two out of every three are divisible by 3, and hence certainly not prime, while in the sequence of values taken by the first polynomial, none are divisible by 3. Thus it seems plausible that the first polynomial will produce values with a higher density of primes than will the second. At the very least, this observation gives little reason to believe that the corresponding diagonals will be equally dense with primes. One should, of course, consider divisibility by primes other than 3. Examining divisibility by 5 as well, remainders upon division by 15 repeat with pattern 1, 11, 14, 10, 14, 11, 1, 14, 5, 4, 11, 11, 4, 5, 14 for the first polynomial, and with pattern 5, 0, 3, 14, 3, 0, 5, 3, 9, 8, 0, 0, 8, 9, 3 for the second, implying that only three out of 15 values in the second sequence are potentially prime (being divisible by neither 3 nor 5), while 12 out of 15 values in the first sequence are potentially prime (since only three are divisible by 5 and none are divisible by 3).

While rigorously-proved results about primes in quadratic sequences are scarce, considerations like those above give rise to a plausible conjecture on the asymptotic density of primes in such sequences, which is described in the next section.