On integers which are representable as sums of large squares

Abstract

We prove that the greatest positive integer that is not expressible as a linear combination with integer coefficients of elements of the set {n 2 , (n+1) 2 , . . .} is asymptotically O(n 2 ), verifying thus a conjecture of Dutch and Rickett. Furthermore we ask a question on the representation of integers as sum of four large squares.2010 Mathematics Subject Classification. 11D07,11P05.

Frobenius Numbers and Automatic Sequences

“…. ., the sequence of squares, Dutch and Rickett [7] proved that G s (n) = o(n 2+ ) for all > 0, and this bound was improved to O(n 2 ) by Moscariello [12].…”

Numerical Semigroups Generated by Quadratic Sequences

“…Only a very small amount of work has appeared on numerical semigroups generated by polynomial sequences, and only for particular instances of polynomials, not for generic polynomials. This includes numerical semigroups generated by three consecutive squares or cubes [8], infinite sequences of squares [11], and sequences of three consecutive triangular numbers or four consecutive tetrahedral numbers [17]. In [3], the authors tantalizingly defined something called a quadratic numerical semigroup; however, the quadratic object in question is an associated algebraic ideal, not a sequence of generators.…”

The Frobenius number for sequences of triangular and tetrahedral numbers

“…. , a n } is part of a "classic" integer sequences: arithmetic and almost arithmetic ( [4,18,11,25]), Fibonacci ( [12]), geometric ( [14]), Mersenne ( [22]), repunit ( [21]), squares and cubes ( [10,13]), Thabit ( [20]), et cetera. For example, in [4] Brauer proves that F(n, n + 1, .…”