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 2019-06-15, 11:45 #1 vasyannyasha     "Vasiliy" Apr 2017 Ukraine 22×3×5 Posts Searching many formulas to one limit Hi guys! This is formula for nth r-gonal pyramidal number $P_n^r= \frac{3n^2 + n^3(r-2) - n(r-5)}{6}$ I have , for example 2
 2019-06-16, 08:43 #2 vasyannyasha     "Vasiliy" Apr 2017 Ukraine 22×3×5 Posts And i forget about searching from some minimal border. How to find nmin without computing previous $P_n^r$? Pre-thx
 2019-06-24, 15:12 #3 lavalamp     Oct 2007 London, UK 130810 Posts What exactly are you searching for?
 2019-06-25, 06:34 #4 vasyannyasha     "Vasiliy" Apr 2017 Ukraine 22·3·5 Posts Im searching for numbers that n-gonal pyramidal number and n-angular number. Cannonball problem for different bases
2019-06-25, 16:37   #5
Dr Sardonicus

Feb 2017
Nowhere

1101111101102 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

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