Linear Relations between Numbers of Terms and First Terms of Sums of Consecutive Squared Integers Equal to Squared Integers

The classical problem of finding all integers a and M such that the sums of M consecutive squared integers a+i2 equal the squared integer s2, where M is the number of terms in the sum, a2 the first term and a >= 1, 0 <= i <= M-1, yields remarkable regular linear features when plotting the values of M as a function of a. These linear features correspond to groupings of pairs of a values for successive same values of M found on either side of straight lines of equation mu M=2a+c, where c is an integer constant and mu a parameter taking some rational values, called allowed values. We find expressions of a and s as a function of M for the allowed values of mu and M and parametric expressions of a, M, and s. Further, Pell equations deduced from the conditions of M are solved to find the allowed values of mu and to provide all solutions in a and M. These results yield new insights into the overall properties of the classical problem of the sums of consecutive squared integers equal to squared integers and allow us to solve this problem completely by providing all solutions in infinite families.