Problems & Puzzles: Puzzles
Puzzle 103. N=a4+b4
is the least number that is the sum of two biquadrates in two different
594+1584 = 1334 + 1344
635318657 is not prime.
Question 1. Find the least and
two more examples where N,
a & c
(if they exist... now I doubt about their existence)
we want that all the four a, b, c & d numbers are primes (of course
that in this case N is even), this is the
2. Find 3 more examples of these.
Question 3. Any shortcut to find them (1 & 2)?
Question 4. A fresh interpretation of the title of the background song by Robin Frost?Refs. pp. 275 & 290-291, "Recreations in The Theory of Numbers", A. H. Beiler, (except for the 4th question)
Chris Nash wrote "N cannot be prime. Because a prime of the form 4x+1 can only be expressed as a sum of two squares in exactly one way. (and fourth powers are of course examples of squares)"
Jean-Charles Meyrignac points out (14/08/2000) that "D.J. Bernstein already explored equations a^4+b^4 = c^4+d^4. You can find the 516 first solutions at http://cr.yp.to/sortedsums/two4.1000000 (detected broken 1/9/01)
In his page Bernstein says also that "218 of the solutions were found previously by Lander, Parkin, Selfridge, and Zajta".
With the help of a little code in Ubasic I (C.R.) analyzed the 516 solutions as DATA statements and found only other prime solution:
620474 + 403514 = 596934 + 467474 = 17472238301875630082
Stuart Gascoigne wrote (2/6/04):
The following numbers are all prime....
157662524204984267584824067340911^4 + 175518862296574361970383530947991^4 = 174460113691778517117959086988941^4 + 159090599632575616653359441298211^4
I took this identity
which was originally published as "Some new results on equal sums of like powers", Simcha Brudno Mathematics of computation volume 25 1969 pp. 877-880. I found it on Jean-Charles Meyrignac's sitehttp://euler.free.fr/identities.htm.
I then tried all values of a<b<1000 and tested the values produced to see if they were prime. I did this as a spreadsheet using Joe Crump's ZZMath Excel addinhttp://www.spacefire.com/numbertheory/default.asp?SubPage=ZMath.htm.
This gave me some values which are 'probable primes'. I then used Primo by Marcel Martinhttp://www.ellipsa.net/ to check they were prime.
The entire computation took maybe one hour and gave me 4 results. In the tradition of these things, I conjecture that there are an infinite number of prime solutions.