Talk:Generalized taxicab number

Candidate solution for Taxicab(5, 2, n)

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)

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)

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)

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

An conjecture

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.

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Generalized taxicab number. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

• Added archive https://web.archive.org/web/20150513224105/http://homepage.ntlworld.com/duncan-moore/taxicab/ to http://homepage.ntlworld.com/duncan-moore/taxicab/

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

• If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
• If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 16:22, 12 October 2017 (UTC)