On the number of representations of an integer by a linear form

Gil Alon, Pete L. Clark

Research output: Contribution to journalArticlepeer-review


Let a1,..., ak be positive integers generating the unit ideal, and j be a residue class modulo L = lcm(a1,..., a k). It is known that the function r(N) that counts solutions to the equation x1a1 + ... + xkak = N in non-negative integers xi is a polynomial when restricted to non-negative integers N = j (mod L). Here we give, in the case of k = 3, exact formulas for these polynomials up to the constant terms, and exact formulas including the constants for q = gcd(a1, a2) · gcd(a1, a3) · gcd(a2, a3) of the L residue classes. The case q = L plays a special role, and it is studied in more detail.

Original languageEnglish
JournalJournal of Integer Sequences
Issue number5
StatePublished - 20 Oct 2005
Externally publishedYes


  • Frobenius problem
  • Pick's theorem
  • Quasi-polynomial
  • Representation numbers


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