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I'd like to find a number that's the difference of fourth powers in three ways or more. I.e.: $$k=a^4-b^4=c^4-d^4=e^4-f^4$$ Is this possible?

There seem to be plenty of examples of differences of fourth powers in two ways. The smallest: $$300783360=133^4−59^4=158^4−134^4$$

I've checked numbers $a,b$ up to ~$10,000$ with a python script with no results.

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    $\begingroup$ The smallest one is $300783360=133^4-59^4=158^4-134^4$. $\endgroup$ – Julián Aguirre Nov 23 '16 at 14:42
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Yes, there are infinitely many. The smallest it seems is,

$$N = 4860992489864937000960$$

and the $3$-way,

$$N =335084^4 - 296668^4= 265076^4 - 93436^4= 264047^4 - 1169^4$$

See the 2007 paper Quartic Diophantine Chains by Choudhry and Wroblewski.

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