Quote:
Originally Posted by vasyannyasha
Im searching for numbers that ngonal pyramidal number and nangular number. Cannonball problem for different bases

For a given r, an rgonal number is of the form
Multiplying by 1/2*(r  2) and adding 1/16*r^2  1/2*r + 1 gives a square y^2.
(Note that when r = 4 you multiply by 1 and add 0).
Thus, for a given r you can write
where the P is the n
^{th} rpyramidal number, which is cubic in n. For any given r, this is an elliptic curve. This gives a mighty bludgeon to use on the problem.
For r = 4, there are elementary proofs that n = 70 is the only n > 1 giving a square value for the cannonball problem.