# Problem of the Month (October 2010)

Jean Lagrange found a set of distinct integers { -15863902, 17798783, 21126338, 49064546, 82221218, 447422978 } with a remarkable property: all 15 sums of 2 distinct elements are square numbers. This is the smallest known 6-set with this property. This raises the question, for an n-set { a1, a2, . . . an }, how many of the sums ai + aj can be square for i≠j? What is the smallest set (in terms of sum) that accomplishes this?

More generally we can ask, for an n-set of distinct integers, how many of the sums of k of the n elements can be square? For example, for the set { –9, 3, 6, 7 } the sum of any 3 elements is square. We can also generalize by wanting the sums to be cubes, triangular numbers, primes, or having some other distinguishing property.

Sums of 2 Elements are Square
nnumber of sumssmallest such setAuthor
21{ –1, 1 }Leo Moser
33{ –4, 4, 5 }Berend van der Zwaag
46{ –94, 95, 130, 194 }Leo Moser
510{ –4878, 4978, 6903, 12978, 31122 }Jean-Louis Nicolas
615{ –15863902, 17798783, 21126338,
49064546, 82221218, 447422978 }
Jean Lagrange

Sums of 2 Elements + 1 are Square
nnumber of sumssmallest such setAuthor
21{ –1, 0 }Bryce Herdt
33{ –2, 1, 2 }Erich Friedman
46{ –32, 32, 67, 256 }Berend van der Zwaag

Sums of 3 Elements are Square
nnumber of sumssmallest such setAuthor
31{ –1, 0, 1 }Bryce Herdt
44{ –9, 3, 6, 7 }Erich Friedman
510{ 26072323311568661931, 43744839742282591947, 118132654413675138222, 186378732807587076747, 519650114814905002347 }Alex Rower

Richard Sabey sent an analysis of sums of n-1 numbers from an n-set adding to n consecutive squares. This can be accomplished for n=1, 2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 18, 20, É

Sums of 2 Elements are Triangular
nnumber of sumssmallest such setAuthor
21{ –1, 1 }Erich Friedman
33{ –1, 1, 2 }Bryce Herdt
46{ –9, 12, 24, 54 }Bryce Herdt

Sums of 2 Elements + 1 are Triangular
nnumber of sumssmallest such setAuthor
21{ –1, 0 }Bryce Herdt
33{ –3, 2, 3 }Erich Friedman
46{ –14, 16, 28, 49 }Erich Friedman

Sums of 3 Elements are Triangular
nnumber of sumssmallest such setAuthor
31{ –1, 0, 1 }Bryce Herdt
44{ –7, 2, 5, 8 }Bryce Herdt

Sums of 2 Elements are Prime
nnumber of sumssmallest such setAuthor
21{ 0, 2 }Bryce Herdt
33{ 0, 2, 3 }Bryce Herdt
45{ 0, 2, 3, 5 }Berend van der Zwaag
57{ –4, –2, 4, 7, 9 }Berend van der Zwaag
610{ –3, –1, 3, 5, 8, 14 }Berend van der Zwaag
714{–6, –4, 6, 8, 11, 23, 35}Berend van der Zwaag
818{–14, –8, 10, 16, 21, 27, 31, 37}Berend van der Zwaag
922{–21, –17, –11, 13, 19, 24, 28, 34, 40}Berend van der Zwaag
1027{–30, –24, –4, 6, 26, 32, 35, 41, 47, 77}Berend van der Zwaag

Sums of 2 Elements + 1 are Prime
nnumber of sumssmallest such setAuthor
21{ 0, 1 }Erich Friedman
33{ 0, 2, 4 }Erich Friedman
46{ –2, 4, 6, 12 }Erich Friedman
510{ –2, 4, 6, 12, 24 }Berend van der Zwaag
615{ –2, 4, 8, 14, 32, 38 }Berend van der Zwaag
721{ –2, 4, 12, 18, 24, 48, 54 }Berend van der Zwaag
828{ –2, 4, 8, 14, 32, 38, 74, 98 }Berend van der Zwaag
936{ –2, 4, 8, 14, 32, 38, 74, 98, 158 }Berend van der Zwaag
1045{ –1, 11, 29, 59, 71, 167, 211, 389, 431, 449 }Berend van der Zwaag

Sums of 3 Elements are Prime
nnumber of sumssmallest such setAuthor
31{ –1, 1, 2 }Erich Friedman
44{ –4, 2, 4, 5 }Berend van der Zwaag
510{ –9, 3, 9, 11, 17 }Erich Friedman
619{ –5, 1, 7, 15, 21, 51 }Erich Friedman
731{ –41, 19, 25, 29, 35, 53, 119}Berend van der Zwaag
849{ –111, 27, 57, 87, 97, 127, 265, 295}Berend van der Zwaag

Sums of 2 Elements are Positive Cubes
nnumber of sumssmallest such setAuthor
21{ –1, 1 }Erich Friedman
33{ –13, 13, 14 }Erich Friedman
46{ –35780, 4693243, 11888132, 70993724 }Bryce Herdt

Sums of 2 Elements + 1 are Positive Cubes
nnumber of sumssmallest such setAuthor
21{ –1, 0 }Bryce Herdt
33{ –4, 3, 4 }Erich Friedman
46{ –27224065, 31881527, 39076416, 43805439 }Bryce Herdt

Sums of 3 Elements are Positive Cubes
nnumber of sumssmallest such setAuthor
31{ –1, 0, 1 }Bryce Herdt
44{ –15, 4, 11, 12 }Erich Friedman

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/13/10.