math.NT — Number Theory

On the ternary pentagonal numbers conjecture

Contributed by Glenn Bruda

Communicated by Guy in 1994, the ternary pentagonal numbers conjecture of Blecksmith and Selfridge asserts that every integer larger than $33066$ is the sum of three positive pentagonal numbers. Prior to this note, it was even unknown whether every sufficiently large integer is the sum of three positive pentagonal numbers. We resolve this in the affirmative, using the landmark work of Duke and Schulze-Pillot on ternary quadratic forms to handle all sufficiently large integers $n$ with $v_3(24n+3)\leq8$, and present an explicit lift to handle the $n$ with $v_3(24n+3)\geq9$.