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I came across the Diophantine equation $5(p^2+q^2+r^2+s^2+t^2)^2 - 7(p^4+q^4+r^4+s^4+t^4) = 90pqrst$ many years ago in the 'Numbers Count' column of the March 1986 issue of 'Personal Computer World' magazine. The column was about Markoff Numbers and Markoff Triples, and simply described this equation as 'a related Diophantine equation'.

It presumably wasn't made up out of thin air, but I've never been able to find any earlier references to it, where it comes from, or if it is indeed related to Markoff numbers. Does anyone recognise it, or have any ideas as to where it might have come from?

(Note: a bit of googling will find the equation in 'Surfing on the Ocean of Numbers' by Henry Ibstedt, but this was in response to the PCW column, not prior to it.)

Edit 2020-12-14: I'm not looking for solutions to this equation, just it's origins.

Duncan Moore
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  • are you looking for solutions? Since the RHS is a quartic in $\max{p,q,r,s,t}$ with the coefficient $-2$, it should be easy to get an upper bound there. – dezdichado Dec 14 '20 at 17:03
  • @dezdichado No, I'm not looking for solutions, just the origins of the equation. I've modified the question to clarify this. – Duncan Moore Dec 14 '20 at 19:00
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    It's a really fascinating problem. In the next issue of that magazine, few solutions of this Diophantine equation is given - $(1,1,1,1,1)$, $(1,1,1,1,2)$, $(2,7,19,47,59)$, $(3,3,4,6,42,87)$ and $(2,13,39,97,99)$. – piepie Dec 15 '20 at 05:59

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You can find it in Hirzebruch and Zagier, "The Atiyah-Singer theorem and elementary number theory" (pg. 159, eqn. (7)).

yoyo
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