Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations
aa r X i v : . [ m a t h . N T ] F e b TRIANGULAR NUMBERS MULTIPLE OF TRIANGULARNUMBERS AND SOLUTIONS OF PELL EQUATIONS
VLADIMIR PLETSER
Abstract.
For all positive non-square integer multiplier k , there is an infinityof multiples of triangular numbers which are also triangular numbers. Witha simple change of variables, these triangular numbers can be found usingsolutions of Pell equations. With some conditions on parities of fundamen-tal solutions of the simple and generalized Pell equations, only odd solutionsof the generalized Pell equation are retained to provide many infinitely solu-tions found on branches corresponding to each of the generalized fundamentalsolutions. General algebraic expressions of fundamental solutions of the Pellequations are found for some values of the multiplier k in function of the closestnatural square. Further, among the expressions of Pell equation solutions, aset of recurrent relations is identical to those found previously without the Pellequation solving method. It is found also that two constants of the problemof multiples of triangular numbers are directly related to the fundamental so-lutions of the simple Pell equation, which is an unexpected result as it meansthat simple Pell equation fundamental solutions in all generality, are related toconstants in recurrent relations of the problem of finding triangular numbersmultiple of other triangular numbers. Introduction
Triangular numbers T t = t ( t +1)2 are figurate numbers with several interest-ing properties and formula (see, e.g., [1, 2]). In this paper, we investigatetriangular numbers T ξ that are multiples of other triangular numbers T t T ξ = kT t (1)Several authors have investigated this Diophantine equation; see, e.g., [3,4, 5, 6, 7, 8, 9]. Further historical accounts can be found in [6]. Recently,Pletser showed [10] that, for non-square integer values of k , the four variables t, ξ, T t and T ξ can be represented by recurrent relations involving a rank r and parameters κ and γ which are respectively the sum and the product ofthe ( r − th and the r th values of t . The rank is being defined as the numberof successive values of t solutions of (1) such that their successive ratios areslowly decreasing without jumps.We only consider solutions of (1) for k > as, for k = 0 and k = 1 , solutionsare trivial, respectively, ξ = 0 and ξ = t for any positive integer t .In this paper, we investigate how to find all solutions to (1) using the methodof resolution of the simple and generalized Pell equations associated to (1).We show that the rank r and parameters κ and γ of recurrent relations can be Key words and phrases.
Triangular Numbers, Multiple of Triangular Numbers, Recurrent Re-lations, Pell Equations, Fundamental Solutions.
Table 1.
Solutions of (1) for k = 3 , k = 3 k = 6 n t n ξ n t n /t n − t n ξ n t n /t n − t n /t n − r and recurrent relations. Section 3 give a short reminder on howto find solutions of Pell equations. In Section 4, Pell equation methodsare applied to find all multiples of triangular numbers that are triangularnumbers. In certain cases, general expressions of fundamental solutions ofthe Pell equations associated to (1) are given for values of the multiplier k in function of the closest natural square values s . Rank and recurrent relations
The Online Encyclopedia of Integer Sequences (OEIS) [11] lists sequencesof solutions of (1) for k = 2 , , , , , . Let us note first that, among allsolutions, ( t , ξ ) = (0 , is always a first solution of (1) for all non-squareinteger value of k .Let’s consider the two cases of k = 3 and k = 6 yielding the successivesolution pairs as shown in Table 1. We indicate also the ratios t n /t n − forboth cases and t n /t n − for k = 6 . It is seen that for k = 3 , the ratio t n /t n − varies between close values, from 5 down to 3.737, while for k = 6 , theratio t n /t n − alternates between values 3 ... 2.385 and 4.667 ... 4.206, whilethe ratio t n /t n − decreases more regularly from 14 to 10.029 (correspondingapproximately to the product of the alternating values of the ratio t n /t n − ).We call rank r the integer value such that t n /t n − r is approximately constantor, better, decreases regularly without jumps (a more precise definition isgiven further). So, here, the case k = 3 has rank r = 1 and the case k = 6 has rank r = 2 .Pletser showed [10] that the rank r is the index of t r and ξ r solutions of (1)such that κ = t r + t r − = ξ r − ξ r − − (2)The rank r is also such that the ratio t r /t r , corrected by the ratio t r − /t r ,is equal to a constant κ + 3 t r − t r − t r = 2 κ + 3 (3) RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS3
For example, for k = 6 and r = 2 , κ = t + t = 3 + 1 = 4 , and κ = ξ − ξ − − − , yielding κ + 3 = 11 .Pletser found [10] four recurrent equations for t n , ξ n , T t n and T ξ n for eachnon-square integer value of kt n = 2 ( κ + 1) t n − r − t n − r + κ (4) ξ n = 2 ( κ + 1) ξ n − r − ξ n − r + κ (5) T t n = (cid:16) κ + 1) − (cid:17) T t n − r − T t n − r + ( T κ − γ ) (6) T ξ n = (cid:16) κ + 1) − (cid:17) T ξ n − r − T ξ n − r + k ( T κ − γ ) (7)where coefficients are functions of two constants κ and γ , respectively thesum (2) and the product γ = t r − t r . Note that these four relations areindependent from the value of k . Pell equations: A Reminder
The Diophantine bivariate quadratic equation X − DY = N, (8)with integers X, Y, D, N and square free D , is called the Pell equation. Sev-eral mathematicians have investigated this equation (see historical accountsin [12, 13, 14, 15, 16]), Treatments and solutions are described in severalclassical text books (see e.g. [17, 19, 18, 20] and references therein). Weremind here some general formulas and how to calculate solutions. Detailscan be found in references.For N = 1 , (8) is called the simple Pell equation x − Dy = 1 (9)This equation admits the obvious trivial solution ( x , y ) = (1 , and infin-itely many solutions given by ( x n , y n ) = (cid:16) x f + √ Dy f (cid:17) n + (cid:16) x f − √ Dy f (cid:17) n , (cid:16) x f + √ Dy f (cid:17) n − (cid:16) x f − √ Dy f (cid:17) n √ D (10)where n are positive integers and ( x f , y f ) is the least solution to (9), i.e.the smallest integer solution different from the trivial solution, x f > , y f > . We call this least solution the fundamental solution. Obviously, havingfound the fundamental solution ( x f , y f ) yields directly three other solutions, ( − x f , y f ) , ( x f , − y f ) , ( − x f , − y f ) .Lagrange devised a method to find the fundamental solution, based on thecontinued fraction expansion of the quadratic irrational √ D , that can be RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS4 summarized as follows. One computes the j th convergent ( p j /q j ) of the con-tinued fraction [ α ; α , . . . , α j , α j +1 , . . . ] of √ D , with α = j √ D k , i.e., thegreatest integer ≤ √ D . This continued fraction becomes periodic after thefollowing term, α j +1 = 2 α if √ D is a quadratic irrational. The recurrencerelations p i = α i p i − + p i − , q i = α i q i − + q i − yield the terms p i and q i of the convergent, with p − = 0 , p − = 1 , q − =1 , q − = 0 . The fundamental solution is then ( x f , y f ) = ( p j , q j ) if j is odd,or ( x f , y f ) = ( p j +1 , q j +1 ) if j is even.For N = 1 , (8) is called the generalized Pell equation, which can have eitherno solution, or one, or several fundamental solutions ( X f i , Y f i ) , with positiveintegers i such that ≤ i ≤ ρ , where ρ is the total number of fundamentalsolutions admitted by (8). All integer solutions, if they exist, are found ondouble infinite branches that can be expressed in terms of the fundamentalsolution(s) ( X f i , Y f i ) and ( − X f i , Y f i ) . Methods to calculate the fundamentalsolution(s) of the generalized Pell equation (see e.g. [17, 19, 20, 21, 22, 23, 24,25] and references therein) are all based on Lagrange’s method of continuedfractions, sometime adapted (see e.g. [26]). The nearest integer continuedfraction method and the Lagrange-Mollin-Matthews method [25] are usedfurther to calculate the fundamental solutions of respectively, the simpleand the generalized Pell equations.Once fundamental solutions are known, the other solutions ( X n , Y n ) of (8)are calculated by X n + √ DY n = ± (cid:16) X f i + √ DY f i (cid:17) (cid:16) x f + √ Dy f (cid:17) n (11)for a proper choice of sign ± [20], yielding respectively, for n = 0 , , (as-suming a + sign), ( X , Y ) = ( X f i , Y f i ) (12) ( X , Y ) = ( X f i x f + DY f i y f , X f i y f + Y f i x f ) (13) ( X , Y ) = (cid:0) X f i (cid:0) x f + Dy f (cid:1) + 2 DY f i x f y f ,Y f i (cid:0) x f + Dy f (cid:1) + 2 X f i x f y f (cid:1) (14)Note that, for each value of n , one can have several (up to ρ ) solutionsdepending on the different values of the generalized fundamental solutions ( X f i , Y f i ) .The other solutions ( X n , Y n ) of (8) can also be represented by recurrencerelations ( X n , Y n ) = ( x f X n − + Dy f Y n − , x f Y n − + y f X n − ) (15)that can also be written as ( X n , Y n ) = (2 x f X n − − X n − , x f Y n − − Y n − ) (16) RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS5 or by Chebyshev polynomials of the first kind T n − ( x f ) and of the secondkind U n − ( x f ) , evaluated at x f (see [27]), ( X n , Y n ) = ( X f i T n − ( x f ) + DY f i y f U n − ( x f ) ,X f i y f U n − ( x f ) + Y f i T n − ( x f )) (17)One notices that the second recurrent relations (16) is similar in form torecurrent relations (4) and (5) found in [10]. Pell Equations and Multiples of Triangular Numbers
Solutions of Pell equations.
For non-square integers k and with thechange of variables ( X, Y ) = (2 ξ + 1 , t + 1) (18)(1) becomes a generalized Pell equation [7, 8], with D = k and N = 1 − k negative, as k > , X − kY = 1 − k (19)and the associated simple Pell equation reads x − ky = 1 (20)Odd solutions ( X, Y ) of (19) provide then pairs ( ξ, t ) , solutions of (1). Fol-lowing the procedure of Section 2, the fundamental solutions of the simpleand generalized Pell equations are calculated and shown in Tables 2 to (4)for non-square k between 2 and 102. The second and third columns give therank r found in [10] and the total number ρ of fundamental solutions of thegeneralized Pell equation.The fourth column shows the single fundamentalsolution of the simple Pell equations; the fifth and sixth columns give thefundamental solutions of the generalized Pell equations, the fifth column forthose solutions with both X f i and Y f i odd or having different parities, whilethe sixth column give those solutions with both X f i and Y f i even (except for k = 56 , see discussion further).From these Tables, we deduce the following.First, the rank of solutions of (1) is equal to, or less than, the total numberof fundamental solutions of the generalized Pell equations, r ≤ ρ , as wasexpected.Second, for all the single fundamental solutions ( x f , y f ) of the simple Pellequation, both x f and y f are of different parities, i.e., one is odd, the othereven (except for some cases of k ≡ mod , where both x f and y f areodd; see further). It is easy to see why: for (20) to hold, the following threeconditions must hold:(C1) x f and y f can not be simultaneously even, whatever the value of k is;(C2) if k is even, x f must necessarily be odd and y f can be either even orodd;(C3) if k is odd, x f and y f must have different parities, one odd and theother even.Third, the sets of fundamental solutions of the generalized Pell equation al-ways include the two fundamental solutions ( X f , Y f ) = (1 , and ( X f , Y f ) = RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS6
Table 2.
Fundamental solutions of simple (19) and generalized(20) Pell equations k r ρ ( x f , y f ) ( X f i , Y f i ) (3 ,
2) (1 , (2 ,
1) (1 , (9 ,
4) ( ± ,
1) (4 , (5 ,
2) ( ± , (8 ,
3) ( ± , (3 ,
1) ( ± ,
10 3 3 (19 ,
6) ( ± , , (9 ,
11 2 2 (10 ,
3) ( ± ,
12 2 2 (7 ,
2) ( ± ,
13 4 6 (649 , ± , , ( ± ,
7) ( ± ,
14 2 2 (15 ,
4) ( ± ,
15 2 2 (4 ,
1) ( ± ,
17 2 3 (33 ,
8) ( ± ,
1) (16 ,
18 2 2 (17 ,
4) ( ± ,
19 3 3 (170 ,
39) ( ± , , (39 ,
20 2 2 (9 ,
2) ( ± ,
21 4 6 (55 ,
12) ( ± , , ( ± ,
3) ( ± ,
22 4 4 (197 ,
42) ( ± , , ( ± ,
23 2 2 (24 ,
5) ( ± ,
24 2 2 (5 ,
1) ( ± ,
26 3 3 (51 ,
10) ( ± , , (25 ,
27 2 2 (26 ,
5) ( ± ,
28 4 4 (127 ,
24) ( ± , , ( ± ,
29 4 6 (9801 , ± , , ( ± ,
11) ( ± ,
30 2 2 (11 ,
2) ( ± ,
31 4 4 (1520 , ± , , ( ± ,
32 2 2 (17 ,
3) ( ± ,
33 2 4 (23 ,
4) ( ± ,
1) ( ± ,
34 2 2 (35 ,
6) ( ± ,
35 2 2 (6 ,
1) ( ± ,
37 2 3 (73 ,
12) ( ± ,
1) (36 ,
38 2 2 (37 ,
6) ( ± ,
39 2 2 (25 ,
4) ( ± ,
40 4 4 (19 ,
3) ( ± , , ( ± ,
41 4 4 (2049 , ± , , ( ± ,
42 2 2 (13 ,
2) ( ± ,
43 4 4 (3482 , ± , , ( ± ,
44 2 2 (199 ,
30) ( ± ,
45 4 6 (161 ,
24) ( ± , , ( ± ,
3) ( ± , RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS7
Table 3.
Fundamental solutions of simple (19) and generalized(20) Pell equations k r ρ ( x f , y f ) ( X f i , Y f i )
46 6 6 (24335 , ± , , ( ± , , ( ± ,
47 2 2 (48 ,
7) ( ± ,
48 2 2 (7 ,
1) ( ± ,
50 3 3 (99 ,
14) ( ± , , (49 ,
51 3 3 (50 ,
7) ( ± , , (35 ,
52 4 4 (649 ,
90) ( ± , , ( ± ,
53 4 6 (66249 , ± , , ( ± ,
29) ( ± ,
54 2 2 (485 ,
66) ( ± ,
55 4 4 (89 ,
12) ( ± , , ( ± ,
56 2 4 (15 ,
2) ( ± ,
1) ( ± ,
57 4 4 (151 ,
20) ( ± , , ( ± ,
58 4 4 (19603 , ± , , ( ± ,
59 2 2 (530 ,
69) ( ± ,
60 2 2 (31 ,
4) ( ± ,
61 8 12 (1766319049 , ( ± , , ( ± , , ( ± , , ± , , ( ± , ± ,
62 2 2 (63 ,
8) ( ± ,
63 2 2 (8 ,
1) ( ± ,
65 2 5 (129 ,
16) ( ± ,
1) ( ± , , (64 ,
66 4 4 (65 ,
8) ( ± , , ( ± ,
67 4 4 (48842 , ± , , ( ± ,
68 2 2 (33 ,
4) ( ± ,
69 4 6 (7775 , ± , , ( ± ,
11) ( ± ,
70 4 4 (251 ,
30) ( ± , , ( ± ,
71 4 4 (3480 , ± , , ( ± ,
72 2 2 (17 ,
2) ( ± ,
73 6 6 (2281249 , ± , , ( ± , , ( ± ,
74 2 2 (3699 , ± ,
75 2 2 (26 ,
3) ( ± ,
76 6 6 (57799 , ± , , ( ± , , ( ± ,
77 4 6 (351 ,
40) ( ± , , ( ± ,
5) ( ± ,
78 4 4 (53 ,
6) ( ± , , ( ± ,
79 2 2 (80 ,
9) ( ± ,
80 2 2 (9 ,
1) ( ± ,
82 3 3 (163 ,
18) ( ± , , (81 ,
83 2 2 (82 ,
9) ( ± ,
84 2 2 (55 ,
6) ( ± ,
85 8 12 (285769 , ( ± , , ( ± , , ( ± , , ± , , ( ± , ± , RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS8
Table 4.
Fundamental solutions of simple (19) and generalized(20) Pell equations k r ρ ( x f , y f ) ( X f i , Y f i )
86 4 4 (10405 , ± , , ( ± ,
87 2 2 (28 ,
3) ( ± ,
88 4 4 (197 ,
21) ( ± , , ( ± ,
89 4 4 (500001 , ± , , ( ± ,
90 2 2 (19 ,
2) ( ± ,
91 6 6 (1574 , ± , , ( ± , , ( ± ,
92 4 4 (1151 , ± , , ( ± ,
93 4 6 (12151 , ± , , ( ± ,
13) ( ± ,
94 4 4 (2143295 , ( ± , , ( ± ,
95 2 2 (39 ,
4) ( ± ,
96 4 4 (49 ,
5) ( ± , , ( ± ,
97 4 8 (62809633 , ( ± , , ( ± ,
59) ( ± , , ± ,
98 2 2 (99 ,
10) ( ± ,
99 2 2 (10 ,
1) ( ± ,
101 2 3 (201 ,
20) ( ± ,
1) (100 ,
102 2 2 (101 ,
10) ( ± , ( − , , which is quite obvious from (19). The only two exceptions are forthe cases k = 2 and . Although ( − , is also a solution to (19) for thesetwo cases, it does not bring a new branch of solutions calculated by (11)to (13) different from the one obtained with (1 , . Therefore, there is onlyone fundamental solution, i.e., ρ = 1 for these two cases. Furthermore, thetwo pairs (1 , − and ( − , − are also solutions of (19), but they do notyield new branches of solutions different from those obtained with ( − , and (1 , .Fourth, all generalized fundamental solutions ( X f i , Y f i ) with i > , i.e., otherthan ( ± , , have both X f i and Y f i odd, except for k = 40 , , , . . . where Y f i is even.Fifth, the generalized fundamental solutions with both X f i and Y f i even areshown separately as they do not bring any solutions to (1), and there are ρ − r such solutions.With the two generalized fundamental solutions (cid:0) X f , , Y f , (cid:1) = ( ± , , onehas from (12) (cid:0) X , , Y , (cid:1) = ( ± , and it yields the two trivial solutions (cid:0) ξ , , t , (cid:1) = (cid:0) ± − , − (cid:1) = (0 , and ( − , of (1). The next generalizedsolution (13) reads (cid:0) X , , Y , (cid:1) = (( ± x f + ky f ) , ( ± y f + x f )) RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS9 yielding, from (18), ( ξ , , t , ) = (cid:18) ± x f + ky f − , ± y f + x f − (cid:19) (21)with both terms integers under the three conditions C1 to C3 above.For other generalized fundamental solutions ( X f i , Y f i ) (with i > ) differentfrom ( ± , , one has from (12) ( X i , Y i ) = ( X f i , Y f i ) , yielding (cid:0) ξ , , t , (cid:1) = (cid:18) X f i − , Y f i − (cid:19) (22)integer solutions of (1) if X f i and Y f i are both odd. The next generalizedsolution (13) reads ( X i , Y i ) = ( X f i x f + kY f i y f , X f i y f + Y f i x f ) , yielding ( ξ i , t i ) = (cid:18) X f i x f + kY f i y f − , X f i y f + Y f i x f − (cid:19) (23)One sees clearly that X f i and Y f i can not be simultaneously even for ξ i and t i to be integers. For X f i and Y f i both odd, the three conditions C1 to C3above ensure that ξ i and t i are integers.For the cases of X f i odd and Y f i even, like for k = 40 and in Tables 2 to4, one has that x f and y f must be simultaneously odd and, by condition C2above, k must be even for (23) to provide integer solutions.Finally, for all single fundamental solutions ( x f , y f ) of the simple Pell equa-tion with both x f and y f odd,they appear for most of the values of k such that k ≡ mod . The exceptions to this are for k = 56 , 72, 112, 184, 240, 248,264, 272, 376, ..., i.e., for some values of k such that k ≡ ± , ±
16 ( mod (but not all), where y f is even. In these cases, one has that k and y f are botheven, then Y f i can not be even for (23) to provide integer solutions. If this isnot the case, i.e., if Y f i is even, then the generalized fundamental solutions ( X f i , Y f i ) must be discarded as it does not provide integer solutions for t in(23).For the general case of k ≡ mod , the fact that y f is not odd can beexplained as follows. As k ≡ mod is not square free, the simple Pellequation (20) can be simplified posing k = c k ′ , with k ′ square free, yielding x − k ′ y ′ = 1 (24)with y ′ = cy . The fundamental solution (cid:16) x f , y ′ f (cid:17) of (24) yields then thefundamental solution ( x f , y f ) = (cid:16) x f , y ′ f c (cid:17) of (20). For example, for k = 8 ,let k ′ = 2 and c = 2 , (24) yields (cid:16) x f , y ′ f (cid:17) = (3 , and (cid:16) x f , y ′ f c (cid:17) = ( x f , y f ) =(3 , . For most of the cases of k such that k ≡ mod , y ′ f is divisible byc such that y ′ f c is odd yielding then y f odd.For the exceptions of some values of k such that k ≡ ± , ±
16 ( mod , thisprocedure does not lead to an odd value of y ′ f c . For example, for k = 56 , RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS10 let k ′ = 14 and c = 2 , yielding (cid:16) x f , y ′ f (cid:17) = (15 , and y f = y ′ f c = 2 . For k = 72 , let k ′ = 2 and c = 6 , yielding (cid:16) x f , y ′ f (cid:17) = (3 , . However, y ′ f is notdivisible by c = 6 and one must consider not the first fundamental solutionof the simple Pell equation for k ′ = 2 , but the second solution given by (10)for n = 2 , yielding ( x , y ′ ) = (17 , that gives y f = y ′ c = 2 and finally ( x f , y f ) = (17 , .Furthermore, for some expressions of k in function of the closest naturalsquare s , one can find general expressions of ( x f , y f ) and ( X f i , Y f i ) in addi-tion to ( ± , (i.e., for i > ) as shown in Table 5. All these expressions caneasily be demonstrated by replacing the appropriate variables in (19) and(20).Note that these general expressions for the fundamental solutions ( x f , y f ) are valid in all generality for the simple Pell equation (20).4.2. First r solutions of (1) for multiple of triangular numbers. Before calculating all solutions of (1) yielding triangular numbers that aremultiple of other triangular numbers, we have to find the first r solutions ( ξ i , t i ) (with ≤ i ≤ r ) of (1), arranged in increasing value order, i.e., ξ = 0 < ξ < . . . < ξ i < . . . < ξ r and , t = 0 < t < . . . < t i < . . . < t r ,and that correspond to the r fundamental solutions ( X f i , Y f i ) of the gener-alized Pell equation (19), with both X f i and Y f i odd or of different parities.The generalized fundamental solutions ( X f , Y f ) = (1 , and ( X f , Y f ) =( − , provide respectively, the solutions ( ξ r , t r ) and ( ξ r − , t r − ) of (1) from(13), yielding successively ( X , Y ) = ( X f x f + kY f y f , X f y f + Y f x f )= ( x f + ky f , y f + x f )( X , Y ) = ( X f x f + kY f y f , X f y f + Y f x f )= ( − x f + ky f , − y f + x f ) and ( ξ r , t r ) = (cid:18) X − , Y − (cid:19) = (cid:18) x f + ky f − , y f + x f − (cid:19) (25) ( ξ r − , t r − ) = (cid:18) X − , Y − (cid:19) = (cid:18) − x f + ky f − , − y f + x f − (cid:19) (26)Then for r > , the next two generalized fundamental solutions ( X f , Y f ) and ( X f , Y f ) = ( − X f , Y f ) yield respectively ( ξ , t ) and ( ξ , t ) . If both RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS11
Table 5.
Expressions of k, s, r, ( x f , y f ) , ( X f i , Y f i ) for i > k s r ( x f , y f ) | ( X f i , Y f i ) s + 1 even 2 (cid:0) ± (cid:0) s + 1 (cid:1) , s (cid:1) | ( − , − ) odd 3 (cid:0) ± (cid:0) s + 1 (cid:1) , s (cid:1) | (cid:0) s , s (cid:1) s + 2 any ( a ) (cid:0) ± (cid:0) s + 1 (cid:1) , s (cid:1) | ( − , − ) s + 4 even ( b ) (cid:16) ± (cid:16) s +22 (cid:17) , s (cid:17) | ( − , − )1 ( mod
4) 4 ( c ) (cid:18) ± ( s +1 ) ( s +4 ) − , s +34 (cid:19) | (cid:16) ± (cid:16) s (cid:16) s − s +42 (cid:17) − (cid:17) , s ( s − + 1 (cid:17) mod (cid:18) ± ( s +1 ) ( s +4 ) − , s +34 (cid:19) | (cid:16) ± (cid:16) s (cid:16) s + s +42 (cid:17) + 1 (cid:17) , s ( s +1)2 + 1 (cid:17) s + 8 0 ( mod (cid:16) ± (cid:16) s + 1 (cid:17) , s (cid:17) | ( − , − )2 ( mod (cid:18) ± s ( s +8 ) + 1 , s (cid:16) s +48 (cid:17)(cid:19) | ( − , − ) s + s any 2 ( ± (2 s + 1) , | ( − , − ) s b ± sσ mod σ ) , ∀ σ odd ≥ (cid:0) ± (cid:0) σs b ± (cid:1) , σ (cid:1) | ( ∗ , ∗ )0 (cid:0) mod σ (cid:1) , ∀ σ even ≥ (cid:0) ± (cid:0) σs b ± (cid:1) , σ (cid:1) ( d ) | ( ∗ , ∗ ) s + s − any ≥ ∗ , ∗ ) | (cid:0) ± (cid:0) s + 2 s − (cid:1) , s + 1 (cid:1) ( e ) s + s − mod >4 ( ∗ , ∗ ) | (cid:16) ± s +4 s − , s + 1 (cid:17) ( f ) mod ( ∗ , ∗ ) | ( − , − )2 ( mod ( ∗ , ∗ ) | (cid:16) ± s − , s − + 1 (cid:17) s + s + 1 1 ( mod (cid:16) ± (cid:16) s +1) + 1 (cid:17) , (cid:16) s − + 1 (cid:17)(cid:17) | (cid:16) ± s +2 s − , s +13 (cid:17) , mod ≥ ∗ , ∗ ) | (cid:0) ± (cid:0) s + 2 s + 1 (cid:1) , s + 1 (cid:1) s + 2 s any 2 ( ± ( s + 1) , | ( − , − ) s + 2 s − any 2 (cid:0) ± (cid:0) s + 2 s (cid:1) , s + 1 (cid:1) | ( − , − ) s + 2 s − mod
3) 2 ( g ) (cid:16) ± s +4 s − , s +1)3 (cid:17) | ( − , − ) s + 2 s − mod (cid:18) ± ( s +1) ( s +2 s − ) , s ( s +2)2 (cid:19) | (cid:16) ± s +3 s − , s +22 (cid:17) mod
4) 4 ( h ) (cid:18) ± ( s +1) ( s +2 s − ) , s ( s +2)2 (cid:19) | (cid:16) ± s + s − , s (cid:17) odd ( i ) (cid:16) ± s +2 s − , s +12 (cid:17) | ( − , − ) s + 2 s − mod
4) 2 ( j ) (cid:16) ± s +2 s − , s +14 (cid:17) | ( − , − )1 ( mod (cid:18) ± s ( s − ) +4 s ( s − ) +18 , ( s − ) ( s +3)8 (cid:19) | (cid:16) ± s +3 s − , s +12 (cid:17) s + (3 s +1)2 odd 2 ( ± (4 s + 3) , | ( − , − )( − , − ) : no solutions exist as r = 2 ; ( ∗ , ∗ ) : no apparent pattern; b ± : plus/minus signindependent from other ± sign; (a) except for k = 51 , ( r = 3 , ); (b) except for k = 40 ( r = 4 ); (c) except for k = 85 ( r = 8 ); (d) except for k = σ − , with σ even ; (e) except for k = 5 , , , . . . ; (f) except for k = 40 ; (g) except for k = 78 ( r = 4 ); (h) except for k = 5 ( r = 2 ); (i) except for k = 96 ( r = 4 ); (j) except for k = 136 ( r = 4 ) RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS12 X f and Y f are odd, then (12) ( n = 0 ) can be used for ( ξ , t ) , yielding ( ξ , t ) = (cid:18) X f − , Y f − (cid:19) (27)Equation (12) could also be used for ( ξ , t ) with ( − X f , Y f ) , but it wouldprovide a negative value for ξ . Instead, we use (13) ( n = 1 ), giving ( ξ , t ) = (cid:18) − X f x f + kY f y f − , − X f y f + Y f x f − (cid:19) (28)The next two generalized fundamental solutions ( X f , Y f ) and ( X f , Y f )= ( − X f , Y f ) yield similarly the next two solutions ( ξ i , t i ) that are put inthe right increasing order.For example, for k = 13 , r = 4 , ( x f , y f ) = (649 , , ( X f i , Y f i ) = ( ± , , ( ± , ,(25) and (26) yield respectively, ( ξ r , t r ) = ( ξ , t ) = (1494 , and ( ξ r − , t r − ) =( ξ , t ) = (845 , ; (27) and (28) yield respectively, ( ξ , t ) = (12 , and ( ξ , t ) = (77 , .Another example, for k = 46 , r = 6 , ( x f , y f ) = (24335 , , ( X f i , Y f i ) =( ± , , ( ± , , ( ± , . With (cid:0) X f , , Y f , (cid:1) = ( ± , , (25) and (26)yield respectively, ( ξ , t ) = (94691 , , ( ξ , t ) = (70356 , . With (cid:0) X f , , Y f , (cid:1) = ( ± , , (12) yields ( ξ , t ) = (cid:16) X f − , Y f − (cid:17) = (23 , and(13) yields ( ξ , t ) = (cid:16) − X f x f + kY f y f − , − X f y f + Y f x f − (cid:17) = (5795 , Finally, with (cid:0) X f , , Y f , (cid:1) = ( ± , , (12) yields ( ξ , t ) = (cid:16) X f − , Y f − (cid:17) =(91 , and (13) yields ( ξ , t ) = (cid:16) − X f x f + kY f y f − , − X f y f + Y f x f − (cid:17) = (1495 , .For the case where Y f i is even, i.e., k = 40 , , , . . . , (25), (26) and(27) cannot be used with (cid:0) X f , , Y f , (cid:1) = ( ± , as both k and Y f i areeven, yielding non-integer solutions for ξ and t . Instead, the other gener-alized fundamental solution have to be used with (13) ( n = 1 ) and (14)( n = 2 ). For example, for k = 40 , r = 4 , ( x f , y f ) = (19 , , ( X f i , Y f i ) =( ± , , ( ± , , (13) yields, first, with ( X f , Y f ) = (11 , , ( X , Y ) =( X f x f + kY f y f , X f y f + Y f x f ) = (449 , , yielding ( ξ , t ) = (224 , ,and second, with ( X f , Y f ) = ( − , , ( X , Y ) = ( X f x f + kY f y f , X f y f + Y f x f ) = (31 , , giving ( ξ , t ) =(15 , . Next, (14) yields, first, with ( X f , Y f ) = (1 , , ( X , Y ) = (cid:16) x f + ky f + 2 kx f y f , x f + ky f + 2 x f y f (cid:17) = (5281 , , yield-ing ( ξ , t ) = (2640 , , and second, with ( X f , Y f ) = ( − , , ( X , Y ) = (cid:16) − (cid:16) x f + ky f (cid:17) + 2 kx f y f , x f + ky f − x f y f (cid:17) = (3839 , , yielding ( ξ , t ) =(1919 , .4.3. All solutions of (1) for multiple of triangular numbers.
Oncethat the first r values of ( ξ i , t i ) have been found, each corresponding to RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS13 one of the r generalized fundamental solutions ( X f i , Y f i ) , the r branches ofinfinitely many other solutions can be found using either :1) the r general solutions (11) (assuming a + sign), yielding ξ n + √ kt n = ξ i + √ kt i + √ k !! (cid:16) x f + √ ky f (cid:17) n − √ k ! (29)where ( ξ i , t i ) must be replaced successively by the r values of ( ξ i , t i ) ; or2) the first recurrence relation (15), yielding ( ξ n , t n ) = (cid:18) ( x f ξ n − r + ky f t n − r ) + (cid:18) x f + ky f − (cid:19) , ( x f t n − r + y f ξ n − r ) + (cid:18) x f + y f − (cid:19)(cid:19) (30)where indices of ξ n − r and t n − r (instead of ξ n − and t n − ) in the right part of(30) refer to the preceding values of ξ and t in the same branch of solutions;or,3) the second recurrence relation (16), yielding ( ξ n , t n ) = (2 x f ξ n − r − ξ n − r + ( x f − , x f t n − r − t n − r + ( x f − (31)where indices of ξ n − r , ξ n − r and t n − r , t n − r (instead of ξ n − , ξ n − and t n − , t n − )in the right part of (31) refer to the preceding and the one before values of ξ and t in the same branch of solutions; or,4) the Chebyshev polynomial solution (17), yielding ( ξ n , t n ) = (cid:18)(cid:18) ξ i + 12 (cid:19) T n − ( x f ) + k (cid:18) t i + 12 (cid:19) y f U n − ( x f ) − , (cid:18) ξ i + 12 (cid:19) y f U n − ( x f ) + (cid:18) t i + 12 (cid:19) T n − ( x f ) − (cid:19) (32)where ( ξ i , t i ) must be replaced successively by the r values of ( ξ i , t i ) .4.4. Relation between Pell equation solutions and recurrent rela-tions.
We can give now a new definition of the rank r introduced in Section2. The rank r is the number of fundamental solutions ( X f i , Y f i ) of the gen-eralized Pell equation (19), with X f i odd and Y f i odd or even (if y f is noteven) , with r ≤ ρ , the total number of generalized solution of (19).Furthermore, we see that the second recurrent relations (31) for both ξ n and t n have x f as the only parameter, and that the two relations are independentfrom the value of k and y f . This fundamental solution x f of the simple Pellequation (20) acts like a constant of the problem for each value of k . Notefurther that summing the expressions of t r and t r − in (25) and (26) yields t r + t r − = x f − . As this sum t r + t r − was already defined in (2), theconstant κ is related to x f κ = x f − (33) RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS14
Furthermore, (33) yields also that y f is related to the difference δ = t r − t r − through the simple Pell equation 20 ( κ + 1) − ky f = 1 , which is verified if y f = κ − t r t r − = ( t r − t r − ) = δ , giving δ = y f (34)for all non-square values of k (except for some values of k such that k ≡ mod , see further). Replacing in the simple Pell equation ( κ + 1) − kδ =1 yields the condition between the sum and the difference of t r and t r − δ = r κ + κk (35)With the exception of k = 56 , 72, 112, 184, 240, 248, 264, 272, 376, ..., i.e.,for some values of k such that k ≡ ± , ±
16 ( mod (for which (34) is valid),the relation (34) is not valid for the other cases of k ≡ mod . In thesecases, δ > y f and one must find the next pair of solutions to the simplePell equation by (10) for n = 2 , i.e., ( x , y ) = (cid:16) x f + ky f , x f y f (cid:17) . Then forthese cases, κ = x f + ky f − (36) δ = 2 x f y f (37)Finally, replacing x f in (31) from 33 yields ( ξ n , t n ) = (2 ( κ + 1) ξ n − r − ξ n − r + κ, κ + 1) t n − r − t n − r + κ ) (38)which are the same recurrent relations given in (4) and (5). Conclusions
We have shown that the problem of finding all triangular numbers thatare multiples of other triangular numbers with non-square integer multiplier k can be solved using solutions of Pell equations with a simple change ofvariables. Only those r fundamental solutions ( X f i , Y f i ) of the generalizedPell equation with X f i odd and Y f i odd or even (if y f is not even) providesolutions to the problem of finding triangular numbers that are multiple ofother triangular numbers. General expressions of fundamental solutions ofthe Pell equations are given for some values of the multiplier k in functionof the closest natural square values s . Many infinitely solutions are thenfound on r branches corresponding to each of the r generalized fundamentalsolutions ( X f i , Y f i ) and these solutions can be found either by a generalrelation involving √ k , or by a first set of recurrent relations, or by a secondset of recurrent relations, or by Chebyshev polynomial solutions. Amongthese, the second set of recurrent relations are found to be the same as thosefound previously without using the Pell equation solving method.Furthermore, the number r of generalized fundamental solutions ( X f i , Y f i ) with X f i odd and Y f i odd or even (if y f is not even) corresponds to the rank ofthese second set recurrent relations. Finally, the two constants κ = t r + t r − RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS15 and δ = t r − t r − are also related to respectively the fundamental solutions x f and y f of the simple Pell equation, as κ = x f − and δ = y f or δ = 2 x f y f .These are an unexpected result as this means that the fundamental solutionsof the simple Pell equation, in all its generality, are related to constants inrecurrent relations of the problem of finding triangular numbers multiple ofother triangular numbers. References [1] E. W. Weisstein, Triangular Number, From MathWorld–A Wolfram WebResource. http://mathworld.wolfram.com/TriangularNumber.html. Lastaccessed 10 February 20201.[2] G. E. Andrews, Number Theory, Dover, New York, 1971.[3] A. Cunningham, Mathematical Questions and Solutions in Continuationof the Mathematical Columns of "the Educational Times"., Volume 75,F. Hodgson, 1901, 87-88.[4] E. de Joncourt, The nature and notable use of the most simple trigonalnumbers. The Hague: Husson, 1762.[5] D. Roegel, A reconstruction of Joncourt’s tableof triangular numbers (1762). LOCOMAT project,https://locomat.loria.fr/joncourt1762/joncourt1762doc.pdf, 2013. Lastaccessed 10 February 2021.[6] L. E. Dickson, Sum of cubes of numbers in arithmetical progression asquare, Ch. XXI in History of the Theory of Numbers, Vol. 2: Diophan-tine Analysis, Dover, New York, 585-588, 2005.[7] J. S. Chahal and H. D’Souza, Some remarks on triangular numbers, in A.D. Pollington and W. Mean, eds.,
Number Theory with an Emphasis onthe Markov Spectrum , Lecture Notes in Pure Math, Dekker, New York,1993, pp. 61–67.[8] T. Breiteig, Quotients of triangular numbers,
The Mathematical Gazette (2015), 243-255.[9] J. S. Chahal, M. Griffin, and N. Priddis, When are multiplesof polygonal numbers again polygonal numbers?, preprint, 2018.http://arxiv.org/abs/1806.07981v2.[10] V. Pletser, Recurrent Relations for Multiple of TriangularNumbers being Triangular Numbers, ArXiv 2101.00998, 2021,http://arxiv.org/abs/2101.00998, last accessed 12 Jan. 2021.[11] N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences,published electronically at https://oeis.org, with t, ξ, T t , T ξ given respec-tively in A053141, A001652, A075528, A029549 (for k = 2 ); A061278,A001571, A076139, A076140 ( k = 3 ); A077259, A077262, A077260, A077261 ( k = 5 ); A077288, A077291, A077289, A077290 ( k = 6 ); A077398, A077401,A077399, A077400 ( k = 7 ); A336623, A336625, A336624, A336626 ( k = 8 );Last accessed 10 February 2021.[12] A. Weil, Number Theory, an Approach through History, Birkhäuser,Boston, 1984. RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS16 [16]
L. E. Dickson, Pell equation, ax + bx + c made a square, Ch. XII inHistory of the Theory of Numbers, Vol. 2: Diophantine Analysis, Dover,New York, 340-400, 2005.[17] T. Nagell, Introduction to Number Theory, Wiley, New York, 195-212,1951.[18] E. W. Weisstein, Pell Equation, from MathWorld–A Wolfram Web Re-source. http://mathworld.wolfram.com/PellEquation.html, last accessed30 January 2021.[19] M. Jacobson, H. Williams, Solving the Pell Equation, Springer-VerlagNew York, 2009.[20] J. P. Robertson, Solving the generalized Pell equation X − DY = N , 31July 2004. See https://nanopdf.com/download/solving-the-generalized-pell-equation-x2-dy-2-n-introduction_pdf, last 30 January 2021.[21] J. L. Lagrange, Solution d’un Problème d’Arithmétique,in Oeuvres de Lagrange, J.-A. Serret (ed.), Vol.1, Gauthier-Villars, Paris, 671–731, 1867. Seehttp://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN308899644&DMDID=DMDLOG_0024&LOGID=LOG_0024&PHYSID=PHYS_0726, lastaccessed 10 February 2021.[22] G. Chrystal, Algebra - An Elementary Text-Book, Part II, 1st ed. Adamand Charles Black, 1900; 2nd ed., New York, Chelsea, 478-488, 1961.[23] R. A. Mollin, Fundamental Number Theory with Applications, CRCPress, New York, 294-307, 1998.[24] K. R. Matthews, The Diophantine Equation x − Dy = N , D > RIANGULAR NUMBERS MULTIPLE OF TRIANGULAR NUMBERS AND SOLUTIONS OF PELL EQUATIONS17
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