Sums of seven octahedral numbers

Abstract

Abstract. We show that for a large class of cubic polynomials f , every sufficiently large number can be written as a sum of seven positive values of f . As a special case, we show that every number greater than e , reducing an open problem due to Pollock to a finite computation.

Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series

“…Descartes initiated the study of octahedral numbers around 1630. In 1850, Pollock conjectured that every positive integer is the sum of at most 7 octahedral numbers, which for finitely many numbers have been proved by Brady [33]. The difference between two consecutive octahedral numbers is a centered square number, i.e., ( ) ( ) ( ) ( ) Further, as in (100), we have…”

“…The Pythagoreans could not have anticipated that figurative numbers would engage after 2000 years leading scholars such as Adrien-Marie Legendre (1752-1833, France), Karl Friedrich Gauss (1777-1855, Germany), Augustin-Louis Cauchy (1789-1857, France), Carl Guslov Jacob Jacobi (1804-1851, Germany), and Waclaw Franciszek Sierpiński (1882( -1969. In 2011, Michel Marie Deza (1939-2016 and Elena Deza (Russia) in their book [3] had given an extensive information about figurative numbers.…”