Talk:Generalized taxicab number

Candidate solution for Taxicab(5, 2, n)[edit]

Hi, I wrote a script today that looked for integer solutions to the equation a^5 + b^5 = c^5 + d^5. The script randomly created candidate solutions, so I had no big hopes of finding anything. Imagine my surprise when the script found that the number 30573545066697597000000000000000000000000000000000000000000000000000000000000000 has a solution for values

a=2611958487637845

b=7883572922413847

c=150101234195497

d=7889857503034283

I checked this in both MATLAB and Google Calculator. If anyone can confirm this, I would be very grateful. Thanks! Johansodling (talk) 16:00, 29 November 2011 (UTC)[reply]

Are you sure your program can handle that number of digits with sufficient precision? Please check for possible typo, as the solution can't be right as stated. a ends with 5 so a^5 also ends with 5. b^5 does not; therefore a^5 + b^5 does not end in 0.

It's also inherently unlikely that the sum of integer cubes would be a multiple of 10^63 (i.e. a number ending in that many zeros). I believe rather that your tools are only working with 17 significant digits, so the identical sum is spurious. – Fayenatic (talk) 23:30, 29 November 2011 (UTC)[reply]

A PARI/GP test shows your numbers are only equal to 16 digits.

? a=2611958487637845;
? b=7883572922413847;
? c=150101234195497;
? d=7889857503034283;
? print(a^5+b^5)
30573545066697597743158152765640234739688459561938126866062924868384225303187132
? print(c^5+d^5)
30573545066697596432224662246738730654780305031754047515464493101149257045408900

PrimeHunter (talk) 02:40, 30 November 2011 (UTC)[reply]

Aah, ok, I thought it was a little too good to be true. Thanks for clarifying. Johansodling (talk) 06:03, 30 November 2011 (UTC)[reply]

An conjecture[edit]

Conjecture: if ±(a_1)^n±(a_2)^n±(a_3)^n±...±(a_m)^n=0 (all a_i are positive integers) and m > 2, then n ≤ 2^m-2. So no positive integer can be written as the sum of two fifth or higher same powers in more than one way, and no fifth or higher powers can be written as the sum of three same powers.

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