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1980] ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS these cannot vanish. 165 Therefore9 So 5 CQ = 0 implies Hence, by Favardfs Theorem, these Rn must be an orthogonal set with respect to some weighting function w1(x) and some range [o9 d] if the integral be considered a Stieltjes integrals If C0 i1 0 S the Rn do not satisfy a three-term recursion formula (unless n = 2) and by applying the contrapositive of the converse, we see that the Rn cannot be an orthogonal set with respect to any weighting function and range. REFERENCES 1. 2. J. Favard. "Sur les polynomes de Tchebicheff." Comptes Rendus Acad. Sei. Paris 200 (1935):2052-2053. G. Szego. Orthogonal Polynomials. Colloquim Publication Volume XXIII. New York: The American Mathematical Society9 1939. ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS INCLUDING THE ALGEBRAIC SUM OF TRIANGULAR NUMBERS Research ANDRZEJ WlgCKOWSKI Fellow of the Alexander von Homboldt at the University of Saarbrucken Foundation The natural number of the form tn = In + 1\ 1/ , 1N V 2 )= I n ( n ) s where n is a natural number, is referred to as the nth triangular number. The aim of this work is to give solutions of some equations and systems of equations in triangular numbers. 1. THE EQUATION ttx + tty = tta It is well known that the equation (1) tx + ty = tz has infinitely many solutions In triangular numbers txs ty9 and t z . amples it follows immediately from the formula: (2) *(2n + l)fc+ t ^tnk + n= For ex- Wl)Hn* We can ask whether there exists a solution of the equation: The answer to this question is positive9 because there exist two solutions: tt„+ tt??= tts3 and **,„, + t t l l , = t t 2 i e . ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS 166 [April The problem of finding all the solutions to equation (1) above will be solved in a subsequent paper. 2. TRIPLES OF TRIANGULAR NUMBERS, THE SUMS OR DIFFERENCES OF ANY TWO OF WHICH ARE ALSO TRIANGULAR NUMBERS The system of three equations: (4) t x ^~ ^ y — U tx + tz = ty + 2tz U9 tV9 — tq , has infinitely many solutions in triangular numbers tx> ty9 tZ9 tU9 tv , and tq. This theorem can be proved by insertion of the following formulas into equations (4): x = n, (5) y = j(tn - 3), z = tn l - 1, 3 u = -^(tn + 1), v = tn, <7 = -2"(tn - 1 ) , where n is a natural number of the form 4/c + 1 or 4fc + 2 for natural k. In particular, putting n = 14, we have: # = 14, z/ = 51, z = 104, w = 53, y = 105, <? = 156. Since tq - tz equations: = t15S tx (D) = t l l s = tW9 - t10k iri t n e + ty ~ tU9 ^x + ^3 ty + tz = we obtain a solution of the system of numbers: tlh + t51 = t53, ^14~^^'104=:^;105S ^y> = £w, t 5 1 + £10if = ^116* We see that there exists a triple of triangular numbers whose sums in pairs are also triangular numbers. The problem of whether there exist three different triangular numbers, the sum of any two of which is a triangular number was formulated by W. Sierpinski [1]. ThdOtiQJMi Suppose that x > y > z; (7.1) (7.2) tx *~ ty = tU9 tx + tM - tv, ty ~> t z — tw I tx "•" ty = tu , tx + tz - tv9 ty (7.3) then each of the systems of equations: - tz = tW9 tx "F ty == tu , tx - t, = tv, ty + tz =-- tw, where x + w9 y + V; where x £ w, y f v; 1980] ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS (7.4) - *v = ~£x + U = ty + tz = £x - ty = *x - tz = h + tz = £ x (7.5) ~tx (7.6) - ty = ^Us s VV where x ± w9 z ± u; t^W9 ^U 5 Uv s ~^W 5 where x ^ w 9 y ^ v 9 z ^ u ; til 9 tx + tz - ty = t y 9 where x + W 9 y + V9 - tz 167 tv9 + *V = t-U 3 £x ~ tz = ^ - tz = ^ W ) £x (7.7) ^ 5 - ty = ^U S ^x - tz = ty s = ty)} t w , where x ^ w ** - t* = ^x (7.8) 9 y ^ v 9 z ^ u ; has infinitely many solutions in triangular numbers tX9ty9 tZ9tU9 tV9andtw. Vfuoofc We prove even more. Each of the following systems of equations has infinitely many solutions in natural numbers x and y. (8.1) "^ 16x +2 + ^16x4-2 + (8.2) (8.3) ^15x + 3 " (8.4) ^12x + 2 ^12x+2 = ^\9x + 2 = ^12x+2 "*" ^ 9cc + 2 = ^16x+2 + ^13x+2 = ^16x+2 + ^12x42 = ^13x42 ~ ^12x+2 = ^16x42 + ^12x42 = ^9x42 = ^9x42 = ^16x42 ~ ^12x42 + = ^ 9 x + 2» ^15x43 + t5x = ty9 ^12x42 + t5x - " ^ i 3 a . + 25 ^20x + 3 ' ty, ^15x+3' ty9 ^20x+35 t b x l ^20x + 35 ty, "^15x + 3 ' ^ 1 3x + 2 ~ Or ^12x42 = ^5x' ^y, +2 + ^9x + 2 = ^12x + 2 + ^9x + 2 = t13x ^15x43' 168 [April ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS ^12a: + 2 (8.5) ^ 9a; + 2 ^15rc+3 (8.6) ^12x+2 + ^16tf + 2 ~ "^13ar + 2 "^16ic+2 + 13a; + 2 » ^12a: + 2 ^13£C + 2 ~ ^12ar + 2 ^52#+2 + (8.7) = ^# » ~ ^20aM = ^5a:» ^ 3 9a; + 2 ^ 6 5a:+3 » '36tf + 2 ty , ^15:r+ 3 (8.8) '9X + 2 ' h5a; + 3 tc The systems of equations (8.1)-(8.8) are, respectively, equivalent to the following equations, for which there exist initial solutions given below: (9.1) (9.2) (9.3) 337a;z + 125a; - yl 425a; 2 175a; + 145a; - y 2 2 + 35a; - y = 3 o, Uo = 0 Z/o = 3 o, 2/o = 3 y = 0, o> yQ 2 + 15a; - y 1408a; + 80a; - y2 2 56x2 y0 0, 200a;2 + 100a?'- y2 (9.8) 0, y = -12, y = -12, (9.5) (9.7) = 3 y = + 110a; - y Six Uo 2 250a: 2 0, 2 (9.4) (9.6) y - -12, y = -12, 2 + 40a; - y2 o, y = o, y = -6, o 0, z/0 = 0 0, y0 = 2 From the theory Of Pell's equation (also referred to as Fermat's equation), it follows that if, simultaneously, k and m are natural numbers, 1, n9 and q are integers, then the product k • m is not a square, and if there exists an initial solution of the equation, (10) kx2 + lx - my2 - ny = q9 in integers x0 and y0, where lxQ + T T ] + lyQ + -y-J f 0, then equation (10) has infinitely many solutions in natural numbers x and y .• Applying this to equations (9.1)-(9.8) we prove that all the systems of equations (8.1)-(8.8) have infinitely many solutions in natural numbers x and y. This theorem is thus proved. 1980] ON SOME SYSTEMS OF DIOPHANTINE EQUATIONS 169 Some years ago, A. Schinzel found the following proof for the statement that there exist infinitely many triples of different triangular numbers for which the sum of any two is a triangular number [private communication from A. Schinzel]. SckLnzeZ1A : ?K.00{ [unpubtl^k^d) x It is well known that the equation 2 - 424z/2 = 1 has infinitely many solutions, where x E 1 (mod 106) [in every solution, we have ±x = 1 (mod 106)]. Putting 25 k = 5y - YQ^(^ - 1) - 1, 1 = |(x - 1) - 50y + 2, we find 3. ^5fc + 4 + ^9fc+6 + ^12fc+9 = h3fe+10' ^12k+9 = ^15k+ll* SYSTEMS OF EQUATIONS INCLUDING THE ALGEBRAIC SUM AND THE PRODUCT OF TRIANGULAR NUMBERS W. Sierpinski [1] has asked whether there exists a pair of triangular numbers such that the sum and the product of these numbers are triangular numbers. We have found some such systems of equations for which there exist one or two solutions in triangular numbers, e.g.: 1. V x "• ty ~tx ~~ ty — t u s ty j — tw , txty (ID (This solution was found by K. Szymiczek [2].) 2. (12) tx "F ty ^9 3. + ^13 = = £# "" ^y tU9 ^16' = = tv s txty = ^9^13 tu, \tx ' *-)~by — t UJ ^90' ^ 9 = ty, txty + ^ ^ 1 3 = ^91' tx/ty — 1 = tWJ ^21'^6 ~ 1 = t^ . (13) ^21 ~ ^6 = ^ 2 0 s 4. (14) ^21^6 = tx ~ ty = t^, txty = 5. t# + t 3 = t u , tyj ~ ty5 txtz ^21 ~ ^6 = h 0 ' and ^98' ^21^6 = ^98 ^63 ~ ^38 = h o ' ^63^38 = ^17285 tx + tz ~ tq, + ^21 s ^35 ^21^35 ^63 + ^219 ^63^219 ty - tz = S i ' = ^539s = ^228' = ^9855° = tu, (continued) 170 ON EULER'S SOLUTION TO A PROBLEM OF DIOPHANTUS—II 5. continued ^29^69 6. [April ^14i+9» ^168^69 ^8 280 * For the system of equations, (16) tx + ty — ~tus txty ~ tv , there exists also the solution: ^505 + ^531 = ^73 3 > = ^50 5^531 ^189980' The author wishes to thank Professor Dr. Andrzej Schinzel for his valuable hints and remarks. REFERENCES 1. 2. W. Sierpinski. Liczby trojkatne (Triangular Numbers). . PZWS Warszawa: Biblioteczka Matematyczna, 1962. (In Polish.) K. Szymiczek. "On Some Diophantine Equations Connected with Triangular Numbers." Sekoja Matematyki (Poland) 4 (1964):17-22. (Zeszyty Naukowe WSP w Katowicach.) ON EULER'S SOLUTION TO A PROBLEM OF DIOPHANTUS—11 JOSEPH ARKIN Turnpike, Spring Valley, V. E. H0GGATT, JR. State University, San Jose, and E. G. STRAUS* of California, Los Angeles, 197 Old Nyack San Jose University NY 10977 CA 95192 CA 90024 1. INTRODUCTION In an earlier paper [1] we considered solutions to a system of equations: x^xj 1 <_ i < j £ n. + 1 = yld ; In this note we look at the generalized problems: (1.1) x^Xj + a = y\. , a + 0. In Section 2 we apply the results of [1] to the solutions of (1.1). Section 3 we consider the following problem: Find n x 2 matrices I ax \&1 a2 ^2 ... • •• In an\ h n / so that a^b- ± ajb-c = ±1 for all 1 £ i < j £ n. In Section 4 we apply the results of Section 3 to get two-parameter families of solutions of (1.1), linear in a, for n = 4. dation *This author's research was supported Grant No. MCS 77-01780. in part by National Science Foun-