Open AccessCongruence properties of indices of triangular numbers multiple of other triangular numbers
Author(s) -
Vladimir Pletser
Publication year - 2021
Publication title -
open journal of mathematical sciences
Language(s) - English
Resource type - Journals
eISSN - 2616-4906
pISSN - 2523-0212
DOI - 10.30538/oms2021.0162
Subject(s) - mathematics , congruence (geometry) , combinatorics , modulo , square (algebra) , integer (computer science) , prime factor , prime power , discrete mathematics , number theory , prime k tuple , congruence relation , multiplier (economics) , almost prime , prime (order theory) , geometry , computer science , programming language , economics , macroeconomics
For any non-square integer multiplier \(k\), there is an infinity of triangular numbers multiple of other triangular numbers. We analyze the congruence properties of indices \(\xi\) of triangular numbers multiple of triangular numbers. Remainders in congruence relations \(\xi\) modulo \(k\) come always in pairs whose sum always equal \((k-1)\), always include 0 and \((k-1)\), and only 0 and \((k-1)\) if \(k\) is prime, or an odd power of a prime, or an even square plus one or an odd square minus one or minus two. If the multiplier \(k\) is twice the triangular number of \(n\), the set of remainders includes also \(n\) and \((n^{2}-1)\) and if \(k\) has integer factors, the set of remainders include multiples of a factor following certain rules. Algebraic expressions are found for remainders in function of \(k\) and its factors, with several exceptions. This approach eliminates those \(\xi\) values not providing solutions.