Which triangular number is a square number

History of the Theory of Numbers, Vol. New York: Dover, pp. Guy, R. Hofstadter, D. New York: Basic Books, Khatri, M.

Student 27 , , Pietenpol, J. Monthly 69 , , Potter, D. Sengupta, D. Teoria Liczb, 3rd ed. Warsaw, Poland: Monografie Matematyczne t.

Silverman, J. A Friendly Introduction to Number Theory. Sloane, N. The Encyclopedia of Integer Sequences. San Diego: Academic Press, Walker, G. Monthly 58 , , Table of Contents Home Introduction. Triangular numbers. A formula for the triangular numbers. The sum of consecutive cubes. By doing so we will see unexpected structures that are inherent in the natural numbers.

This is number Count them Since 10 can be pictured in this way, we call 10 a triangular number. So the next triangular number is 3. The number we add to the previous triangular number is called the gnomon NOH-mon. We added the gnomon 2 to 1. And so the first four triangular numbers are 1, 3, 6, Each one is the sum of consecutive numbers. Problem 1. Write the first ten triangular numbers. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" "Reload".

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Show that if this happens, then we have:. I've looked at the book's solution and the solution is as follows:. This allows us to complete the square and also makes the equation "look good". As the other answers have explained, the multiplication by 4 is to make things neater. The series of 'square triangulars' can be found by finding all a-b pairs which fulfil the above equation. The first few pairs are: 1,1 ; 3,2 ; 7,5 ; 17, These yeild the square triangular numbers: 1, 36, , But it is necessary to write the formula describing their solutions through solving Pell's equation:.

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