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Basically what the title says. Is there a name(or research papers) for diophantine equations of the form $a_0^p+a_1^p+a_2^p+...+a_{p-1}^p = 1$ where $p$ is a prime and the $a_i$'$s$ are integers? If not, is there a name for a similar diophantine equation? I ask because i have a feeling it has already been well studied.

Thanks in advance

eagle I
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    Perhaps you could start by searching for individual small values of $p$, e g https://en.wikipedia.org/wiki/Sums_of_three_cubes gives a parametric solution (K. Mahler in 1936).

    You might get a better response by showing what you have tried, where this originates, and any numerical solutions you have.

     – Old Peter Sep 08 '23 at 18:05

1 Answers1

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The more general Diophantine equation,

$$a_0^p+a_1^p+a_2^p+...+a_n^p = b^p$$

is called "equal sums of like powers". A good website is http://euler.free.fr/ Yours is just the special case $b=1$ and $n=p-1$. That special case has solutions only for $p = 3,5,7$ for now.

I. p = 3

There are infinitely many examples, the most famous is Ramanujan’s taxicab number $1729$,

$$9^3+10^3 = 12^3+1 =1729$$

a special case of the well-known identity,

$$(9 t^4)^3 + (1 - 9 t^3)^3 = (9 t^4 - 3 t)^3 + 1$$

though there are infinitely many such parameterizations. One can also use Pell equations as in this post.

II. p = 5

There are only eight known examples, the smallest found by Lander and Parkin back in the 1960s,

$$38^5 + 47^5 + 123^5 - (89^5 + 118^5) = 1$$

III. p = 7

There are only two known examples found by Nuutti Kuosa back in 2002, one of which is,

$$130^7 + 1031^7 + 1951^7 + 2787^7 - (1146^7 + 1348^7 + 2816^7) = 1$$

IV. p = 9

There is no known example yet. The closest so far is $b=2$ by Jaroslaw Wroblewski also in 2002,

$$1634^9 + 1564^9 + 1067^9 + 1062^9 + 762^9 - (1735^9 + 819^9 + 305^9 + 228^9) = 2^9$$

P.S. So, yes, these kinds of Diophantine equations and variants thereof have been studied. But not that well-studied as results are old and scanty (and which is a shame considering the computing power now at our disposal).

Tito Piezas III
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