Congruence Properties of Indices of Triangular Numbers Multiple of Other Triangular Numbers

It is known that, for any positive non-square integer multiplier $k$, there is an infinity of multiples of triangular numbers which are triangular numbers. We analyze the congruence properties of the indices $\xi$ of triangular numbers that are multiples of other triangular numbers. We show that the remainders in the congruence relations of $\xi$ modulo k come always in pairs whose sum always equal $\left(k-1\right)$, always include 0 and $\left(k-1\right)$, and only 0 and $\left(k-1\right)$ 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 $\left(n^{2}-1\right)$ and if $k$ has integer factors, the set of remainders include multiples of a factor following certain rules. Finally, algebraic expressions are found for remainders in function of $k$ and its factors. Several exceptions are noticed and superseding rules exist between various rules and expressions of remainders. This approach allows to eliminate in numerical searches those $\left(k-\upsilon\right)$ values of $\xi_{i}$ that are known not to provide solutions, where $\upsilon$ is the even number of remainders. The gain is typically in the order of $k/\upsilon$, with $\upsilon\ll k$ for large values of $k$.


Introduction
Triangular numbers T t = t(t+1) 2 are one of the figurate numbers enjoying many properties; see, e.g., [1,2] for relations and formulas. Triangular numbers T ξ that are multiples of other triangular number T t (1.1) T ξ = kT t are investigated. Only solutions for k > 1 are considered as the cases k = 0 and k = 1 yield respectively ξ = 0 and ξ = t, ∀t. Accounts of previous attempts to characterize these triangular numbers multiple of other triangular numbers can be found in [3,4,5,6,7,8,9]. Recently, Pletser showed [9] that, for non-square integer values of k, there are infinitely many solutions that can be represented simply by recurrent relations of the four variables t, ξ, T t and T ξ , involving a rank r and parameters κ and γ, which are respectively the sum and the product of the (r − 1) th and the r th values of t. The rank r is being defined as the number of successive values of t solutions of (1.1) such that their successive ratios are slowly decreasing without jumps. In this paper, we present a method based on the congruent properties of ξ (mod k), searching for expressions of the remainders in function of k or of its factors. This approach accelerates the numerical search of the values of t n and ξ n that solve (1.1), as it eliminates values of ξ that are known not to provide solutions to (1.1). The gain is typically in the order of k/υ where υ is the number of remainders, which is usually such that υ ≪ k. Table 1. OEIS [10] references of sequences of integer solutions of (1.1) for k = 2, 3, 5, 6, 7, 8

Rank and Recurrent Equations
Sequences of solutions of (1.1) are known for k = 2, 3, 5, 6, 7, 8 and are listed in the Online Encyclopedia of Integer Sequences (OEIS) [10], with references given in Table 1. Among all solutions, t = 0 is always a first solution of (1.1) for all non-square integer value of k, yielding ξ = 0. Let's consider the two cases of k = 2 and k = 7 yielding the successive solution pairs as shown in Table 2. We indicate also the ratios t n /t n−1 for both cases and t n /t n−2 for k = 7. It is seen that for k = 2, the ratio t n /t n−1 varies between close values, from 7 down to 5.829, while for k = 7, the ratio t n /t n−1 alternates between values 2.5 ... 2.216 and 7.8 ... 7.23, while the ratio t n /t n−2 decreases regularly from 19.5 to 16.023 (corresponding approximately to the product of the alternating values of the ratio t n /t n−1 ). We call rank r the integer value such that t n /t n−r is approximately constant or, better, decreases regularly without jumps (a more precise definition is given further). So, here, the case k = 2 has rank r = 1 and the case k = 7 has rank r = 2.
In [9],we showed that the rank r is the index of t r and ξ r solutions of (1.1) such that and that the ratio t 2r /t r , corrected by the ratio t r−1 /t r , is equal to a constant 2κ+3 For example, for k = 7 and r = 2, (2.1) and (2.2) yield respectively, κ = 7 and 2κ + 3 = 17. Four recurrent equations for t n , ξ n , T tn and T ξn are given in [9] for each non-square integer value of k t n = 2 (κ + 1) t n−r − t n−2r + κ (2.3) where coefficients are functions of two constants κ and γ, respectively the sum κ and the product γ = t r−1 t r of the first two sequential values of t r and t r−1 . Note that the first three relations (2.3) to (2.5) are independent from the value of k.

Congruence of ξ modulo k
We use the following notations: for A, B, C ∈ Z, B < C, C > 1, A ≡ B (mod C) means that ∃D ∈ Z such that A = DC + B, where B and C are called respectively the remainder and the modulus. To search numerically for the values of t n and ξ n that solve (1.1), one can use the congruent properties of ξ (mod k) given in the following propositions. In other words, we search in the following propositions for expressions of the remainders in function of k or of its factors. Proposition 1. For ∀s, k ∈ Z + , k non-square, ∃ξ, µ, υ, i, j ∈ Z + , such that if ξ i are solutions of (1.1), then for ξ i ≡ µ j (mod k) with 1 ≤ j ≤ υ, the number υ of remainders is always even, υ ≡ 0 (mod 2), the remainders come in pairs whose sum is always equal to (k − 1), and the sum of all remainders is always equal to the product of (k − 1) and the number of remainder pairs, Proof. Let s, i, j, k, ξ, µ, υ, α, β ∈ Z + , k non-square, and ξ i solutions of (1.1). Rewriting (1.1) as T ti = T ξi /k, for T ti to be integer, k must divide exactly T ξi = ξ i (ξ i + 1) /2, i.e., among all possibilities, k divides either ξ i or (ξ i + 1), yielding two possible solutions ξ i ≡ 0 (mod k) or ξ i ≡ −1 (mod k), i.e. υ = 2 and the set of µ j includes {0, (k − 1)}. This means that ξ i are always congruent to either 0 or (k − 1) modulo k for all non-square values of k. Furthermore, if some ξ i are congruent to α modulo k, then other ξ i are also congruent to β modulo k with β = (k − α − 1). As ξ i ≡ α (mod k), then ξ i (ξ i + 1) /2 ≡ (α (α + 1) /2) (mod k) and replacing α by In this case, υ = 4 and the set of µ j includes, but not necessarily limits to, Note that in some cases, υ > 4, as for k = 66, 70, 78, 105, ... , ν = 8. However, in some other cases, υ = 2 only and the set of µ j contains only {0, (k − 1)}, as shown in the next proposition. In this proposition, several rules (R) are given constraining the congruence characteristics of ξ i .
(R3): If k = s 2 + 1 with s even, the rank r is always r = 2 [11], and the only two sets of solutions are as can be easily shown. For t 1 , forming which is the triangular number of ξ 1 . One obtains similarly ξ 2 from t 2 . These two relations (3.1) and (3.2) show respectively that ξ 1 is congruent to 0 modulo k and , and the only two sets of solutions are as can be easily demonstrated as above. These two relations (3.3) and (3.4) show that ξ 1 and ξ 2 are congruent respectively to (k − 1) and 0 modulo k.
, and the only two sets of solutions are as can easily be shown as above. These two relations (3.5) and (3.6) show that ξ 1 and ξ 2 are congruent respectively to (k − 1) and 0 modulo k.
There are other cases of interest as shown in the next two Propositions Proposition 3. For ∀n ∈ Z + , ∃k, ξ, µ < k, i, j ∈ Z + , k non-square, such that if ξ i are solutions of (1.1) with ξ i ≡ µ j (mod k), and (R6) if k is twice a triangular number k = n (n + 1) = 2T n , then the set of µ j includes 0, n, n 2 − 1 , (k − 1) , with 1 ≤ j ≤ υ.
Finally, this last proposition gives a general expression of the congruence ξ i (mod k) for most cases to find the remainders µ j other than 0 and (k − 1).

Proposition 4.
For ∀n > 1 ∈ Z + , ∃k, f, ξ, ν < n < k, µ < k, m < n, i, j ∈ Z + , k non-square, let ξ i be solutions of (1.1) with ξ i ≡ µ j (mod k), let f be a factor of k such that f = k/n with f ≡ ν (mod n) and k ≡ νn mod n 2 , then the set of µ j in- where m is an integer multiplier of f in the congruence relation and such that m < n/2 or m < (n + 1) /2 for n being even or odd respectively, and 1 ≤ j ≤ υ.
Note that 11 of these 16 values of k are multiple of 6, the others are 2 mod 6 and 5 mod 6 for, respectively three and two cases. One notices as well, that generally, Ra and Exy supersede Ezt with x < z and t < y, except for k = 60 and 120.

Conclusions
We have shown that, for indices ξ of triangular numbers multiples of other triangular numbers, the remainders in the congruence relations of ξ 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 a 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 multiple of a factor following certain rules. Finally, algebraic expressions are found for remainders in function of k and its factors. Several exceptions are noticed as well and it appears that there are superseding rules between the various rules and expressions. This approach allows to eliminate in numerical searches those (k − υ) values of ξ i that are known not to provide solutions of (1.1), where υ is the even number of remainders. The gain is typically in the order of k/υ, with υ ≪ k for large values of k.