Information-theoretic security

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In mathematics, the generalized taxicab number Taxicab(k, j, n) is the smallest number which can be expressed as the sum of j kth positive powers in n different ways. For k = 3 and j = 2, they coincide with Taxicab numbers.

It has been shown by Euler that

$\mathrm{T}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{b}(4,2,2)=635318657=59^4+158^4=133^4+134^4.$

However, Taxicab(5, 2, n) is not known for any n ≥ 2; no positive integer is known which can be written as the sum of two fifth powers in more than one way.

It can be easily verified on a home computer using a simple brute force search that the Taxicab(5, 2, 2) problem has no solutions with { a, b, c, d } all less than 1,000. A more in-depth search shows the same is true for all combinations up to 4,000. A lower bound on the solution is

$\mathrm{T}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{b}(5,2,2)>1,024,000,000,000,000,000=1.024*10^{18}.$

See also

• Cabtaxi number

External links

• Walter Schneider:Taxicab numbers