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

Alon, Gil; Clark, Pete L.

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

Alon, Gil ; Clark, Pete L.

Journal of integer sequences, Tome 8 (2005) no. 5.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Résumé

Summary: Let

a_{1}, \dots, a_{k}

be positive integers generating the unit ideal, and

j

be a residue class modulo

L = lcm (a_{1}, \dots, a_{k})

. It is known that the function

r (N)

that counts solutions to the equation

x_{1} a_{1} + \dots + x_{k} a_{k} = N

in non-negative integers

x_{i}

is a polynomial when restricted to non-negative integers

N \equiv 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 (a_{1}, a_{2}) \cdot gcd (a_{1}, a_{3}) \cdot gcd (a_{2}, a_{3})

of the

L

residue classes. The case

q = L

plays a special role, and it is studied in more detail.

Classification : 05A15, 52C07
Mots-clés : Frobenius problem, quasi-polynomial, representation numbers, pick's theorem

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Alon, Gil; Clark, Pete L. On the number of representations of an integer by a linear form. Journal of integer sequences, Tome 8 (2005) no. 5. https://geodesic-test.mathdoc.fr/item/JIS_2005__8_5_a2/