Divisible by 12: a^3 + b^3 = c^2

[Pfoertner, Hugo]

-----Original Message-----
From: Ed Pegg Jr [mailto:edp at wolfram.com] 
Sent: Wednesday, October 20, 2004 23:17
Cc: seqfan
Subject: Divisible by 12: a^3 + b^3 = c^2


Kirk Bresniker notes that for a^3 + b^3 = c^2, if a and b
are prime, then for a<b<100000, c has a factor of 12:

11^3    + 37^3    = (2 * 2 * 3  * 19)^2
1801^3  + 56999^3 = (2 * 2 * 3  * 5 * 7  * 32401)^2
2137^3  + 8663^3  = (2 * 2 * 3  * 3 * 5  * 4513)^2
6637^3  + 86291^3 = (2 * 2 * 3  * 2 * 2  * 11 * 31 * 1549)^2
8929^3  + 28703^3 = (2 * 2 * 3  * 2 * 2  * 7  * 37 * 397)^2
44111^3 + 48817^3 = (2 * 2 * 3  * 2 * 2  * 7  * 11 * 3847)^2
57241^3 + 87959^3 = (2 * 2 * 3  * 5 * 11 * 44641)^2

Seems like it should be extendable and provable.

--Ed Pegg Jr

Ed, SeqFans,

with a little program I tested the range c^2 <= 4*10^18 and got

      a       b          c       c/12

     11      37        228         19
   1801   56999   13608420    1134035
   2137    8663     812340      67695
   3889  367823  223078944   18589912
   6637   86291   25354032    2112836
   8929   28703    4935504     411292
  16931  133597   48880608    4073384
  38557 1586531 1998370272  166530856
  44111   48817   14218512    1184876
  54083  334717  194057640   16171470
  57241   87959   29463060    2455255
 127717  552011  412662012   34388501
 151477  246011  135516108   11293009
 457511  786697  763311948   63609329
 571019  735781  764539380   63711615
 765983 1154017 1409359200  117446600

No counterexmaple for the divisibility by 12 was found.

If we sort the table by the c column

     11      37        228         19
   2137    8663     812340      67695
   8929   28703    4935504     411292
   1801   56999   13608420    1134035
  44111   48817   14218512    1184876
   6637   86291   25354032    2112836
  57241   87959   29463060    2455255
  16931  133597   48880608    4073384
 151477  246011  135516108   11293009
  54083  334717  194057640   16171470
   3889  367823  223078944   18589912
 127717  552011  412662012   34388501
 457511  786697  763311948   63609329
 571019  735781  764539380   63711615
 765983 1154017 1409359200  117446600
  38557 1586531 1998370272  166530856

we get a sequence

Numbers n such that 12*n^2 can be expressed as the sum of the cubes of two
primes:

19 67695 411292 1134035 1184876 2112836 2455255 4073384 11293009
16171470 18589912 34388501 63609329 63711615 117446600 166530856

Best regards to all and a warm welcome back to Neil!

Hugo