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Taxicab$(n,2,2)$ is the smallest number expressible as the sum of 2 $n$th powers in 2 different ways. I believe only the first 4 are known:

Taxicab$(1,2,2) = 4 = 1^1 + 3^1 = 2^1 + 2^1$,

Taxicab$(2,2,2) = 50 = 1^2 + 7^2 = 5^2 + 5^2$,

Taxicab$(3,2,2) = 1729 = 1^3 + 12^3 = 9^3 + 10^3$,

Taxicab$(4,2,2)= 635318657 = 59^4 + 158^4 = 133^4 + 134^4$.

Where can I find current research on this sequence? Are there some good online resources for investigating these numbers?

Supware
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  • Start with the references from Wikipedia. Have you read the book by Guy on this? – Dietrich Burde Feb 15 '18 at 13:59
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    Searching OEIS for 635318657 quickly leads to http://oeis.org/A046881. One comment gives that an exhaustive search found $\operatorname{Taxicab}(5, 2, 2) > 2^{96}$. – Travis Willse Feb 15 '18 at 14:14
  • The Wikipedia references seem to focus on Taxicab$(n,j,k)$ with $j,k \neq 2$. I haven't looked into any books; I will if nothing appropriate turns up online – Supware Feb 15 '18 at 14:15
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    Years ago, I looked for a solution of the (6,2,2) equation, up to 3.2E+28, without finding any result. – Old Peter Feb 16 '18 at 15:27
  • Power sums $(a,b,c)$ where $a=b+c$ are ultra-rare: few examples for $(4,1,3)$ (of the form $A^4=B^4+C^4+D^4$) and slightly more for $(4,2,2)$ (of the form $A^4+B^4=C^4+D^4$) - see http://mathworld.wolfram.com/DiophantineEquation4thPowers.html . And no power sum $(a,b,c)$ where $a>b+c$ is known yet. So the chance to find such $(5,2,2)$-sum (i.e. Taxicab(5,2,2) ) is very small. – Oleg567 Feb 16 '18 at 20:04

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