Literature Review for $a^5 + b^5 = c^5 + d^5$

Question

I am preparing a paper on a^5 + b^5 = c^5 + d^5, and am having considerable difficulty in the literature review.

I don't see anything on StackExchange. According to Wolfram's Mathworld, a book (Guy, 1994, page 140) includes the assertion that this equation has been checked to 10^26 with no solution. (http://mathworld.wolfram.com/DiophantineEquation5thPowers.html) However, I don't have a copy of that book and it isn't free online.

I was a little surprised that 1994 is the most current reference on this problem that has been studied since at least 300 - 400 years ago.

Q: Is there any information available on this problem?

Answer 1

Guy's book (Unsolved Problems in Number Theory, Second Ed.) was discussing the generalisation with equal sums of $m$ $s$th powers. The relevant paragraph (Chapter D, Section 1, page 140) reads:

Parametric solutions are known for equal sums of equal numbers of like powers, $$\sum_{i=1}^ma_i^s=\sum_{i=1}^mb_i^s$$ with $a_i>0$, $b_1>0$, for $2\leq s\leq4$ and $m=2$ and for $s=5,6$ and $m=3$. Can a solution be found for $s=7$ and $m=4$? For $s=5,m=2$, it is not known if there is any nontrivial solution of $a^5+b^5=c^5+d^5$. Dick Lehmer once thought that there might be a solution with a sum of about $25$ decimal digits, but a search by Blair Kelly III yielded no nontrivial solution with sum $\leq 1.02\times10^{26}$.

The book does not cite Lehmer, but Kelly's paper is cited as: John B. Kelly, Two equal sums of three squares with equal products, Amer. Math. Monthly, 98(1991) 527-529; MR 92j:11025.