Notes on Explorations 395The question of finding distinct sets with equal sums of powers pops
up persistently in the literature. The most detailed treatment appears in
Albert Gloden, Mehrgrudige Gleichungen, (Noordhoff, Groningen, 1944)
MR 8, 441f.
A discussion of the problem, in particular version (d), is found in E.M.
Wright, Prouhet’s 1851 solution of the Tarry-Escott problem of 1910,
Amer. Math. Monthly 66 (1959), 199-201. To divide the set of numbers up
to d”+l into d sets with the desired property, write each number in the set
to base d, sum the digits and classify according to the congruence of this
sum modulo d.
A history of the problem can be found in L.E. Dickson, History of the
Theory of Numbers, (Washington, 1920; reprint, Chelsea, 1952) Vol. II,
Chapt. 24. There is also a section (pages 328-332) on the problem in G.H.
Hardy & E.M. Wright, An Introduction to the Theory of Numbers, (Oxford,
4th ed., 1960).
A recent reference in which the problem plays a role is Gerald Myerson,
How small can a sum of roots be? Amer. Math. Monthly 93 (1986), 457-
459.
Other references related to the topic are
J. Chernick, Ideal solutions of the Tarry-Escott problem
Amer. Math. Monthly 44 (1937), 626-633.
L.E. Dickson, Introduction to the Theory of Numbers
(1929), pages 55-58.
H.L. Dorwart & O.E. Brown, The Tarry-Escott problem
Amer. Math. Monthly 44 (1937), 613-626.
A. Gloden, Two theorems on multi-degree equalities
Amer. Math. Monthly 53 (1946), 205.
A. Gloden, Parametric solutions of two multi-degreed equalities
Amer. Math. Monthly 55 (1948), 86-88.
A. Gloden, Normal trigrade and cyclic quadrilaterals with integral sides
and diagonals
Amer. Math. Monthly 53 (1951), 244-247.
Maurice Kraitchik, Mathematical Recreations
(Norton, 1942; Dover), 79
D.H. Lehmer, The Tarry-Escott problem
Scripta Math. 13 (1947), 37-41.
Joseph S. Madachy, Mathematics on Vacation
(Scribner’s, 1966), pages 173-175.
J.B. Roberts, A curious sequence of signs
Amer. Math. Monthly 64 (1957), 317-322.