Problems & Puzzles: Puzzles

Puzzle 103. N=a4+b4 = c4+d4
("Three Sheets In The Ocean, One Foot In The Sunset, And You"  by Robin Frost, 1986)

635318657 is the least number that is the sum of two biquadrates in two different ways.

635318657= 594+1584 = 1334 + 1344

But 635318657 is not prime.

Question 1. Find the least and two more examples where N, a & c are prime (if they exist... now I doubt about their existence)

If 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 least example:

3262811042 = 74+2394 = 1574+2274

Question 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)

Solution

Question 1

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)"

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Question 2

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)

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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 + 46747= 17472238301875630082

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Stuart Gascoigne wrote (2/6/04):

The following numbers are all prime....

157662524204984267584824067340911^4 + 175518862296574361970383530947991^4 = 174460113691778517117959086988941^4 + 159090599632575616653359441298211^4

269758686329508376469851929750719^4 + 534065979590887060090455886663777^4 =
374021128974810647471091942797893^4 + 508918261456711931134576407608941^4

1679539956802461427023806692932554869^4+2864939822128245005298014063613916133^4= 2735636962517662175684582548040182073^4+2096549864621014042130013452441703601^4

I took this identity

f2(a,b)=-a^13+a^12*b+a^11*b^2+5*a^10*b^3+6*a^9*b^4-12*a^8*b^5-4*a^7*b^6+7*a^6*b^7-3*a^5*b^8-3*a^4*b^9+4*a^3*b^10+2*a^2*b^11-a*b^12+b^13

f2(a,b)^4+f2(b,-a)^4=f2(a,-b)^4+f2(b,a)^4

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 site .

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 addin

This gave me some values which are 'probable primes'. I then used Primo by Marcel Martin 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.

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