[math-fun] News on Taxicab and Cabtaxi numbers
The known information on Taxicab and Cabtaxi numbers goes up to Taxicab(6) and Cabtaxi(9). Here are new upper bounds, up to Taxicab(12) and Cabtaxi(20): http://cboyer.club.fr/Taxicab.htm A paper is in preparation. Christian.
Interesting that someone is still pursuing this. The recent history of Taxicab numbers is interesting. In 1991 Rosenstiel, Dardis & Rosensteil found Taxicab(4) and three other 4-way sums of 2 cubes by in effect filling the memory of their computer with numbers of the form x^3 + y^3 and searching for quadruplicate values. In 1998 I wrote a heap-based algorithm with which I extended RD&R's results and found Taxicab(5) = 48988659276962496. At the time D. Bernstein was using similar methods to attack other Diophantine equations, and, unaware of my work, rediscovered Taxicab(5) some months later. In my paper on Taxicab(5), I gave a simple method for searching (n+1)-way sums from n-way sums, specifically, if x = a^3 + b^3 is an n-way sum, then k^3 x = (ka)^3 + (kb)^3 is an n-way sum as well, so we can search for (n+1)-way sums by starting with an n-way sum x and testing k^3 x for k = 2,3,4,... to see if it has an additional sum not of the form (ka)^3 + (kb)^3, making it an (n+1)-way sum. In 2002, Rathbun took advantage of my neglect and, applying my own methods to my own value of Taxicab(5), discovered that 79^3 * Taxicab(5) = 24153319581254312065344 was a 6-way sum and hence an upper bound on Taxicab(6). Subsequent investigations have increased the confidence that Rathbun's number is in fact Taxicab(6). Hoisted by my own petard. Christian, just curious, are you applying similar methods to come up with your upper bounds? ----- Original Message ----- From: "Christian Boyer" <cboyer@club-internet.fr> To: "'math-fun'" <math-fun@mailman.xmission.com> Sent: Monday, December 11, 2006 5:43 AM Subject: [math-fun] News on Taxicab and Cabtaxi numbers
The known information on Taxicab and Cabtaxi numbers goes up to Taxicab(6) and Cabtaxi(9).
Here are new upper bounds, up to Taxicab(12) and Cabtaxi(20): http://cboyer.club.fr/Taxicab.htm
A paper is in preparation.
Christian.
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David Wilson wrote:
Christian, just curious, are you applying similar methods to come up with your upper bounds?
Yes, similar methods. And I think that the same methods were already used by André Gérardin in 1908 (without computer, of course) for his 3-way solution, which was an upper bound of the true Taxicab(3) found by John Leech 50 years later (with an EDSAC computer). Your very interesting work will be of course mentioned in my future paper! 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 : lundi 11 décembre 2006 19:54 À : math-fun Objet : Re: [math-fun] News on Taxicab and Cabtaxi numbers Interesting that someone is still pursuing this. The recent history of Taxicab numbers is interesting. In 1991 Rosenstiel, Dardis & Rosensteil found Taxicab(4) and three other 4-way sums of 2 cubes by in effect filling the memory of their computer with numbers of the form x^3 + y^3 and searching for quadruplicate values. In 1998 I wrote a heap-based algorithm with which I extended RD&R's results and found Taxicab(5) = 48988659276962496. At the time D. Bernstein was using similar methods to attack other Diophantine equations, and, unaware of my work, rediscovered Taxicab(5) some months later. In my paper on Taxicab(5), I gave a simple method for searching (n+1)-way sums from n-way sums, specifically, if x = a^3 + b^3 is an n-way sum, then k^3 x = (ka)^3 + (kb)^3 is an n-way sum as well, so we can search for (n+1)-way sums by starting with an n-way sum x and testing k^3 x for k = 2,3,4,... to see if it has an additional sum not of the form (ka)^3 + (kb)^3, making it an (n+1)-way sum. In 2002, Rathbun took advantage of my neglect and, applying my own methods to my own value of Taxicab(5), discovered that 79^3 * Taxicab(5) = 24153319581254312065344 was a 6-way sum and hence an upper bound on Taxicab(6). Subsequent investigations have increased the confidence that Rathbun's number is in fact Taxicab(6). Hoisted by my own petard. Christian, just curious, are you applying similar methods to come up with your upper bounds? ----- Original Message ----- From: "Christian Boyer" <cboyer@club-internet.fr> To: "'math-fun'" <math-fun@mailman.xmission.com> Sent: Monday, December 11, 2006 5:43 AM Subject: [math-fun] News on Taxicab and Cabtaxi numbers
The known information on Taxicab and Cabtaxi numbers goes up to Taxicab(6) and Cabtaxi(9).
Here are new upper bounds, up to Taxicab(12) and Cabtaxi(20): http://cboyer.club.fr/Taxicab.htm
A paper is in preparation.
Christian.
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3:41 PM
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participants (2)
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Christian Boyer -
David Wilson