A number which is simultaneously square and triangular. Let
denote the th
triangular number and the th square number, then a number
which is both triangular and square satisfies the equation , or
(Conway and Guy 1996). The first few solutions are , (17, 12), (99, 70), (577, 408), .... These give
the solutions ,
(8, 6), (49, 35), (288, 204), ... (OEIS A001108
and A001109), corresponding to the triangular
square numbers 1, 36, 1225, 41616, 1413721, 48024900, ... (OEIS A001110;
Pietenpol 1962). In 1730, Euler showed that there are an infinite number of such
solutions (Dickson 2005).
The general formula for a square triangular number
is ,
where
is the th
convergent to the continued fraction of
(Ball and Coxeter 1987, p. 59; Conway and Guy 1996). The first few are
Allen, B. M. "Squares as Triangular Numbers." Scripta Math.20, 213-214, 1954.Ball, W. W. R.
and Coxeter, H. S. M. Mathematical
Recreations and Essays, 13th ed. New York: Dover, 1987.Conway,
J. H. and Guy, R. K. The
Book of Numbers. New York: Springer-Verlag, pp. 203-205, 1996.Dickson,
L. E. History
of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover,
pp. 10, 16, and 27, 2005.Guy, R. K. "Sums of Squares"
and "Figurate Numbers." §C20 and §D3 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 136-138
and 147-150, 1994.Hofstadter, D. R. Fluid
Concepts & Creative Analogies: Computer Models of the Fundamental Mechanisms
of Thought. New York: Basic Books, 1996.Khatri, M. N. "Triangular
Numbers Which are Also Squares." Math. Student27, 55-56, 1959.Pietenpol,
J. L. "Square Triangular Numbers." Problem E 1473. Amer. Math.
Monthly69, 168-169, 1962.Potter, D. C. D. "Triangular
Square Numbers." Math. Gaz.56, 109-110, 1972.Sengupta,
D. "Digits in Triangular Squares." College Math. J.30, 31,
1999.Sierpiński, W. Teoria Liczb, 3rd ed. Warsaw, Poland:
Monografie Matematyczne t. 19, p. 517, 1950.Sierpiński, W.
"Sur les nombres triangulaires carrés." Pub. Faculté d'Électrotechnique
l'Université Belgrade, No. 65, 1-4, 1961.Sierpiński,
W. "Sur les nombres triangulaires carrés." Bull. Soc. Royale
Sciences Liège, 30 ann., 189-194, 1961.Silverman, J. H.
A
Friendly Introduction to Number Theory. Englewood Cliffs, NJ: Prentice Hall,
1996.Sloane, N. J. A. Sequences A000129/M1413,
A001333/M2665, A001108/M4536,
A001109/M4217, and A001110/M5259
in "The On-Line Encyclopedia of Integer Sequences."Sloane,
N. J. A. and Plouffe, S. The
Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.Walker,
G. W. "Triangular Squares." Problem E 954. Amer. Math. Monthly58,
568, 1951.