The pentagonal theorem of sixty-three and generalizations of Cauchy’s lemma

Abstract

Abstract In this paper, we consider the solvability over non-negative integers of certain Diophantine equations coming from representations of integers as sums of pentagonal numbers (counting the number of dots in a regular pentagon). We study a general method to obtain generalized versions of Cauchy’s lemma. Using this, we show the “pentagonal theorem of 63”, which states that a sum of pentagonal numbers represents every non-negative integer if and only if it represents the integers 1 , 2 , 3 , 4 , 6 , 7 , 8 , 9 , 11 , 13 , 14 , 17 , 18 , 19 , 23 , 28 , 31 , 33 , 34 , 39 , 42 , 63 . 1,~{}2,~{}3,~{}4,~{}6,~{}7,~{}8,~{}9,~{}11,~{}13,~{}14,~{}17,~{}18,~{}19,~{}23% ,~{}28,~{}31,~{}33,~{}34,~{}39,~{}42,~{}63. We further show that these integers form a unique minimal universality criterion set.