396 Notes on Explorations

J.B. Roberts, Splitting consecutive integers into classes with equal power
sums
Amer. Math. Monthly 71 (1964), 25-37.

J.S. Vidger, Consecutive integers having equal sums of squares
Math. Mag. 38 (1965), 35-42.

T.N. Sinha, On the Tarry-Escott problem
Amer. Math. Monthly 73 (1966), 280-285.
E.M. Wright, On Tarry’s problem
Quart. J. Math. Oxford (1) 6 (1935), 261-267.

Number curiosities
Crux Mathematicorum (Eureka) 2 (1976), 62.
Here are some problems references:

963 Crzlx Mathematicorum 11 (1985), 292-296.

El504 Amer. Math. Monthly 69 (1962), 165, 924.

E.3. See for example, pages 111-129 of C.S. Liu, Introduction to Combina-
torial Mathematics, (McGraw-Hill, 1968), or pages 162-171 of Alan Tucker,
Applied Combinatorics, (Wiley, 1980).

E.4. An account of geometric methods for solving quadratic equations ap-
pears on pages 59-62 and pages 69-70 of Howard Eves, An Introduction to
the History of Mathematics, (5th edition; Saunders, 1983). There is a fairly
detailed discussion of the Euclidean technique of application of areas as well
as an exercise on the approaches of Carlyle and von Staudt (1798-1867).
For visual methods for equations of higher degree, consult T.R. Running,
Graphical solutions of cubic, quartic and quintic, Amer. Math. Monthly 50
(1943), 170-173.

E.5. It is impossible to find four distinct square integers in arithmetic
progression. The problem is discussed in Crux Mathematicorum 8 (1982),
281-282 (Problem 677). Proofs appear in W. Sierpinski, Elementary Theory
of Numbers, (New York, 1964), pp. 74-75; and L.J. Mordell, Diophantine
Equations, (Academic Press), pp. 20-22. For a history of the problem, con-
sult L.E. Dickson, History of the Theory of Numbers, (Washington, 1920;
reprint, Chelsea, 1952), Vol. II, p. 440.
The quadratic 60t2 - 60t + 1 takes successive square values at t = -2,
-1, 0, 1, 2, 3. This can be discovered by noting that the second order
differences of the sequence 361, 121, 1, 1, 121,361 are constant (see Section
2.1, Exploration E.18; Exercise 7.1.17).
We can try to generalize this example to have the quadratic take the
successive integer values

22 y2 x2 1 1 x2 yz %2.

The conditions for constant second order differences are

3x2-g =2 (1)