Square triangular number
In mathematics, a square triangular number (or triangular square number) is a number which is both a triangular number and a perfect square. There are an infinite number of square triangular numbers; the first few are 0, 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, 55420693056, 1882672131025 (sequence A001110 in OEIS).
Write Nk for the kth square triangular number, and write sk and tk for the sides of the corresponding square and triangle, so that
We define the triangular root of a triangular number to be . From this definition and the quadratic formula, we get Therefore, is triangular if and only if is square, and naturally is square and triangular if and only if is square, i. e., there are numbers and such that . This is an instance of the Pell equation, with n=8. All Pell equations have the trivial solution (1,0), for any n; this solution is called the zeroth, and indexed as . If we let denote the k'th non-trivial solution to any Pell equation for a particular n, it can be shown by the method of descent that and . Hence there are an infinity of solutions to any Pell equation for which there is one non-trivial one, which holds whenever n is not a square. The first non-trivial solution when n=8 is easy to find: it is (3,1). A solution to the Pell equation for n=8 yields a square triangular number and its square and triangular roots as follows: and Hence, the first square triangular number, derived from (3,1), is 1 (how exciting!), and the next, derived from (17,6) (=6×(3,1)-(1,0)), is 36.
Other equivalent formulas (obtained by expanding this formula) that may be convenient include
The corresponding explicit formulas for sk and tk are :13
With a bit of algebra this becomes
and then letting x = 2t + 1 and y = 2s, we get the Diophantine equation
and therefore all solutions are given by
There are many identities about the Pell numbers, and these translate into identities about the square triangular numbers.
A. V. Sylwester gave a short proof that there are an infinity of square triangular numbers, to wit:
If the triangular number n(n+1)/2 is square, then so is the larger triangular number
We know this result has to be a square, because it is a product of three squares: 2^2 (by the exponent), (n(n+1))/2 (the n'th triangular number, by proof assumption), and the (2n+1)^2 (by the exponent). The product of any numbers that are squares is naturally going to result in another square, which can best be proven by geometrically visualizing the multiplication as the multiplying of a NxN box by an MxM box, which is done by placing one MxM box inside each cell of the NxN box, naturally producing another square result.
The generating function for the square triangular numbers is:
Relations with other numerical puzzles
|This section does not cite any references or sources. (December 2014)|
The square triangular numbers are connected with several other mathematical puzzles, due to their relation with the Pell equation .
First, the quasi-isosceles Pythagorean triples. These are triples such that and also . The first few are (0,1,1), (3,4,5), and (20,21,29). They are connected to the square triangular numbers, or rather their square and triangular roots and , by and . We can derive (a, b, c) from (s, t) by and
Second is Dudeney's problem of the professor's house number. The professor lives on a long street with house numbers that start at one and go up to a number to be determined by the solver, not skipping any integers. The professor's house number is such that the sum of all the numbers less than his is exactly equal to the sum of all the numbers greater than his. For example, . If the professor's house number is and the street has houses for the same k, then the problem is solved. A variation on this problem has the professor living on the odd-numbered side of the street, and has only the odd-numbered houses added up to create the two matching totals. Example: . This variation is solved by letting the professor live on house (numbered ) on the side of the street having a total of houses (the last numbered ), where is a quasi-isosceles Pythagorean triple, described above.
Third is Martin Gardner's problem of the Jones sisters: Mr. Jones has some daughters, and if you meet any two of them at random on the street, the probability that both have blue eyes is exactly one-half. How many daughters does Mr. Jones have, and how many of them have blue eyes? Mathematically, we are looking for two numbers and such that . One solution is and , i. e., there are 21 daughters and 15 of them have blue eyes. This problem can also be stated in another form as the problem of Socrates and the hemlock: Socrates has just been convicted of slandering the gods, and instead of just executing him, the Greek tribunal decides to let the gods determine his fate. So they arrange some number of glasses in a circle, and put water in some of them and hemlock in the others. The number of glasses, and the number with hemlock, are such that when Socrates drinks the contents of two of them, he has exactly a 50-50 chance of survival. The only difference between this formulation of the problem and Gardner's one is that b is now the number of glasses with water, not hemlock. The solution to these problems is given by and , where is a quasi-isosceles Pythagorean triple.
As becomes larger, the ratio approaches and the ratio of successive square triangular numbers approaches . The table below shows values of between 0 and 7.
- Dickson, Leonard Eugene (1999) . History of the Theory of Numbers 2. Providence: American Mathematical Society. p. 16. ISBN 978-0-8218-1935-7.
- Euler, Leonhard (1813). "Regula facilis problemata Diophantea per numeros integros expedite resolvendi (An easy rule for Diophantine problems which are to be resolved quickly by integral numbers)". Memoires de l'academie des sciences de St.-Petersbourg (in Latin) 4: 3–17. Retrieved 2009-05-11.
According to the records, it was presented to the St. Petersburg Academy on May 4, 1778.
- Barbeau, Edward (2003). Pell's Equation. Problem Books in Mathematics. New York: Springer. pp. 16–17. ISBN 978-0-387-95529-2. Retrieved 2009-05-10.
- Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford University Press. p. 210. ISBN 0-19-853171-0.
- Weisstein, Eric W., "Square Triangular Number", MathWorld.
- Ball, W. W. Rouse; Coxeter, H. S. M. (1987). Mathematical Recreations and Essays. New York: Dover Publications. p. 59. ISBN 978-0-486-25357-2.
- Pietenpol, J. L.; A. V. Sylwester; Erwin Just; R. M Warten (February 1962). "Elementary Problems and Solutions: E 1473, Square Triangular Numbers". American Mathematical Monthly (Mathematical Association of America) 69 (2): 168–169. ISSN 0002-9890. JSTOR 2312558.
- Plouffe, Simon (August 1992). "1031 Generating Functions" (PDF). University of Quebec, Laboratoire de combinatoire et d'informatique mathématique. p. A.129. Retrieved 2009-05-11.