Some implications of the 1907 Hurwitz formula II

When/Where:

September 16, 2014, at 1:55pm, in LIT 368

Abstract:

Let \(s(n)\) denote \(\#\) of representations of integer \(n\) by \(x^2+y^2+z^2\). Let \(n\) be odd positive integer. In my first talk I showed that \(s(n^2)\geq 6n\) with equality if and only if all prime divisors of \(n\) are congruent to \(1 \pmod{4}\). I employ the above inequality and a special case of the Jacobi triple product identity to prove certain conjectures of Kaplansky. In particular, I will show that \(4x^2 + 9y^2 + 32z^2 +4xy \) represents, exclusively, all positive integers not of the form

• \(4^a(4m+2), a=0,1 \)
• \(4^a(8m+3), a=0,1 \)
• \(4^a(8m+5), a=0,1 \)
• \(4^a(8m+7), a\geq 0,\)
• or \(M^2 \)

where \(a,m,M\) are non-negative integers and \(M\) is generated by \(1\) and primes congruent to \(1 \pmod{4} \).