Puzzle 12: Solution

Q     What does each of the following pairs share (besides having the same sum)?
(3, 3, 12) = (1, 8, 9)
(4, 8, 9, 21) = (3, 7, 14, 18)
(4, 7, 21, 36) = (1, 12, 27, 28)
(5, 7, 10, 14, 27) = (3, 6, 15, 18, 21)
(5, 85, 85, 169, 425) = (13, 17, 125, 289, 325)
(17, 21, 24, 48, 54, 238) = (3, 4, 14, 119, 126, 136)

A     You may have noticed that for each pair the products contain the same prime factors, though they aren't equal. What is equal is the product of xx:

33 33 1212 = 11 88 99
44 88 99 2121 = 33 77 1414 1818
44 77 2121 3636 = 11 1212 2727 2828
55 76 1010 1414 2727 = 33 66 1515 1818 2121
55 8585 8585 169169 425425 = 1313 1717 125125 289289 325325
1717 2121 2424 4848 5454 238238 = 33 44 1414 119119 126126 136136

This was suggested by one of the exhibits at the Ramanujan museum at SASTRA University in Kumbakonam, India, which I saw at the Dec.2003 conference where the museum was inaugurated. Ramanujan found several examples, including (3,3,12)=(1,8,9). I used LLL (the algorithm of Lenstra, Lenstra, and Lovasz for lattice basis reduction) to find the rest. The example (5,85,85,169,425) = (13,17,125,289,325) illustrates that all numbers involved can be odd (and indeed coprime to 6). The examples (5,27,28,70) = (6,15,49,60), (1,6,8,20,30) = (2,2,12,24,25), and (with all numbers odd) (3,21,21,25,63) = (5,7,27,45,49) have equal sums, products of x, and products of xx.

This use of LLL is the same trick I had used to find “subset anagrams” — see Alex Healy's note “Finding Anagrams Via Lattice Reduction” here, and Alex's interactive implementation. My talk at the AMS/MAA joint meeting in 2003 about this and other novel applications of lattice reduction was written up in Science by Barry Cipra (Vol. 299 (2003), 650-651).