Let a, n be positive integers that are relatively prime. We say that a/n can be represented as an Egyptian fraction of length k if there exist positive integers m_1 , . . . , m_k such that a/n=1/m_1+ ··· +1/m_ k . Let A k (n) be the number of solutions a to this equation. In this article, we give a formula for A 2 (p) and a parametrization for Egyptian fractions of length 3, which allows us to give bounds to A 3 (n), to f a (n) = #{(m 1 , m 2 , m 3 ) : a/n=1/m_1+ ··· +1/m_ k}, and finally to F (n) = #{(a, m 1 , m 2 , m 3 ) : a/n=a/n=1/m_1+ ··· +1/m_ k}.
Luca, F., Banderier, C., Alexis Gomez Ruiz, C., Pappalardi, F., Treviño, E. (2021). ON EGYPTIAN FRACTIONS OF LENGTH 3. REVISTA DE LA UNION MATEMATICA ARGENTINA, 62(1), 257-274 [10.33044/REVUMA.1798].
ON EGYPTIAN FRACTIONS OF LENGTH 3
Francesco PappalardiMembro del Collaboration Group
;
2021-01-01
Abstract
Let a, n be positive integers that are relatively prime. We say that a/n can be represented as an Egyptian fraction of length k if there exist positive integers m_1 , . . . , m_k such that a/n=1/m_1+ ··· +1/m_ k . Let A k (n) be the number of solutions a to this equation. In this article, we give a formula for A 2 (p) and a parametrization for Egyptian fractions of length 3, which allows us to give bounds to A 3 (n), to f a (n) = #{(m 1 , m 2 , m 3 ) : a/n=1/m_1+ ··· +1/m_ k}, and finally to F (n) = #{(a, m 1 , m 2 , m 3 ) : a/n=a/n=1/m_1+ ··· +1/m_ k}.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
