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2019-06-25, 16:37   #5
Dr Sardonicus

Feb 2017
Nowhere

22·821 Posts

Quote:
 Originally Posted by vasyannyasha Im searching for numbers that n-gonal pyramidal number and n-angular number. Cannonball problem for different bases
For a given r, an r-gonal number is of the form

$p_r^{n} \;=\; n((r-2)*n-(r-4))/2$

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

$(r - 2)P_r^{n}/2 \; + \; \frac{r^{2}}{16}\;-\;\frac{r}{2}\;+\;1\;=\;y^{2}$

where the P is the nth r-pyramidal 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.

Last fiddled with by Dr Sardonicus on 2019-06-25 at 16:39 Reason: xingif posty