[math-fun] Taxicab and prime numbers
It seems that nobody knows a solution of this Taxicab equation with prime numbers (p, q, r, s, t, u): (1) p^3 + q^3 = r^3 + s^3 = t^3 + u^3 I tried also the equivalent Cabtaxi problem. This new problem should be easier, because allowing both sums and differences. But, I am also unable to get a solution of: (2) p^3 +/- q^3 = r^3 +/- s^3 = t^3 +/- u^3 It is "irritating", because the standard solutions of (2) using prime and non-prime are small and easy to find, the smallest being: 728 = 6^3 + 8^3 = 9^3 - 1^3 = 12^3 - 10^3 Why??? Are (1) and (2) impossible? Any idea? With only (p, q, r, s), the equations have solutions, the smallest being: (3) p^3 + q^3 = r^3 + s^3 6507811154 = 31^3 + 1867^3 = 397^3 + 1861^3 12906787894 = 593^3 + 2333^3 = 1787^3 + 1931^3 (4) p^3 +/- q^3 = r^3 +/- s^3 62540982 = 397^3 - 31^3 = 1867^3 - 1861^3 (based on the first above solution of (3)!) 105161238 = 193^3 + 461^3 = 709^3 - 631^3 Christian.
Worst! It seems impossible to find solutions of the more general equations: (5) p^3 + q^3 = r^3 + s^3 = m^3 + n^3 or (6) p^3 +/- q^3 = r^3 +/- s^3 = m^3 +/- n^3 with p, q, r, s primes m, n integers (primes or non-primes) Why? Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : mardi 9 janvier 2007 20:12 À : math-fun@mailman.xmission.com Objet : [math-fun] Taxicab and prime numbers It seems that nobody knows a solution of this Taxicab equation with prime numbers (p, q, r, s, t, u): (1) p^3 + q^3 = r^3 + s^3 = t^3 + u^3 I tried also the equivalent Cabtaxi problem. This new problem should be easier, because allowing both sums and differences. But, I am also unable to get a solution of: (2) p^3 +/- q^3 = r^3 +/- s^3 = t^3 +/- u^3 It is "irritating", because the standard solutions of (2) using prime and non-prime are small and easy to find, the smallest being: 728 = 6^3 + 8^3 = 9^3 - 1^3 = 12^3 - 10^3 Why??? Are (1) and (2) impossible? Any idea? With only (p, q, r, s), the equations have solutions, the smallest being: (3) p^3 + q^3 = r^3 + s^3 6507811154 = 31^3 + 1867^3 = 397^3 + 1861^3 12906787894 = 593^3 + 2333^3 = 1787^3 + 1931^3 (4) p^3 +/- q^3 = r^3 +/- s^3 62540982 = 397^3 - 31^3 = 1867^3 - 1861^3 (based on the first above solution of (3)!) 105161238 = 193^3 + 461^3 = 709^3 - 631^3 Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
My conjecture is that in a 3-way sum, at most two of the cubes can be cubes of primes. ----- Original Message ----- From: "Christian Boyer" <cboyer@club-internet.fr> To: "'math-fun'" <math-fun@mailman.xmission.com> Sent: Wednesday, January 10, 2007 3:08 AM Subject: RE: [math-fun] Taxicab and prime numbers Worst! It seems impossible to find solutions of the more general equations: (5) p^3 + q^3 = r^3 + s^3 = m^3 + n^3 or (6) p^3 +/- q^3 = r^3 +/- s^3 = m^3 +/- n^3 with p, q, r, s primes m, n integers (primes or non-primes) Why? Christian.
Counter-example of your conjecture, with 4 cubes of primes: 41^3 - 2^3 = 89^3 - 86^3 = 40^3 + 17^3 41, 2, 89 and 17 are primes. Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de David Wilson Envoyé : mercredi 10 janvier 2007 09:41 À : math-fun Objet : Re: [math-fun] Taxicab and prime numbers My conjecture is that in a 3-way sum, at most two of the cubes can be cubes of primes. ----- Original Message ----- From: "Christian Boyer" <cboyer@club-internet.fr> To: "'math-fun'" <math-fun@mailman.xmission.com> Sent: Wednesday, January 10, 2007 3:08 AM Subject: RE: [math-fun] Taxicab and prime numbers Worst! It seems impossible to find solutions of the more general equations: (5) p^3 + q^3 = r^3 + s^3 = m^3 + n^3 or (6) p^3 +/- q^3 = r^3 +/- s^3 = m^3 +/- n^3 with p, q, r, s primes m, n integers (primes or non-primes) Why? Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Puzzle 386, asked by Carlos Rivera: http://www.primepuzzles.net/ Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : mercredi 10 janvier 2007 09:09 À : 'math-fun' Objet : RE: [math-fun] Taxicab and prime numbers Worst! It seems impossible to find solutions of the more general equations: (5) p^3 + q^3 = r^3 + s^3 = m^3 + n^3 or (6) p^3 +/- q^3 = r^3 +/- s^3 = m^3 +/- n^3 with p, q, r, s primes m, n integers (primes or non-primes) Why? Christian. -----Message d'origine----- De : math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com [mailto:math-fun-bounces+cboyer=club-internet.fr@mailman.xmission.com] De la part de Christian Boyer Envoyé : mardi 9 janvier 2007 20:12 À : math-fun@mailman.xmission.com Objet : [math-fun] Taxicab and prime numbers It seems that nobody knows a solution of this Taxicab equation with prime numbers (p, q, r, s, t, u): (1) p^3 + q^3 = r^3 + s^3 = t^3 + u^3 I tried also the equivalent Cabtaxi problem. This new problem should be easier, because allowing both sums and differences. But, I am also unable to get a solution of: (2) p^3 +/- q^3 = r^3 +/- s^3 = t^3 +/- u^3 It is "irritating", because the standard solutions of (2) using prime and non-prime are small and easy to find, the smallest being: 728 = 6^3 + 8^3 = 9^3 - 1^3 = 12^3 - 10^3 Why??? Are (1) and (2) impossible? Any idea? With only (p, q, r, s), the equations have solutions, the smallest being: (3) p^3 + q^3 = r^3 + s^3 6507811154 = 31^3 + 1867^3 = 397^3 + 1861^3 12906787894 = 593^3 + 2333^3 = 1787^3 + 1931^3 (4) p^3 +/- q^3 = r^3 +/- s^3 62540982 = 397^3 - 31^3 = 1867^3 - 1861^3 (based on the first above solution of (3)!) 105161238 = 193^3 + 461^3 = 709^3 - 631^3 Christian. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (2)
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Christian Boyer -
David Wilson