Equal sums of powers of the terms of arithmetic progressions of equal lengths

This paper gives a method of obtaining two arithmetic progressions of equal arbitrary length and consisting entirely of positive integers such that th e sums of either the squares or the cubes or the fourth powers of the terms of the two arithmetic progressions are Equal. It is further shown that an arbitrarily large number of such arithmetic progressions can be obtained su ch that the sums of the squares of the terms of all these arithmetic progre ssions are equal.