A sequence a in a group G is called an arithmetic progression of order h = 0, 1, 2 ..., if [D.sup.h+1] a = 0.
It is well known that the sequence a in G is an arithmetic progression of order h if and only if there exists a polynomial p(n) in n, with coefficients in G and of degree less than or equal to h, such that p(n) = [a.sub.n], for every n = 0, 1, 2 ...; that is, there are [[gamma].sub.h], [[gamma].sub.h-1], ..., [[gamma].sub.1], [[gamma].sub.0] [member of] G, which depend only on a, such that, for every n = 0, 1, 2, ...,
Thus we conclude that whenever both the first column and the first row are arithmetic progressions (2), the "fixed" sum
If in a six by six array in which all rows are translations of the first row, and the numbers in both the first row and column form arithmetic progressions, then the sum of any six numbers of the array, no two of which are in the same row or column, is six times the arithmetic mean of the numbers in the upper left and lower right corners of the array.
Progressive primes A computer search revealed an arithmetic progression
consisting of 23 primes, the longest such sequence yet found (sciencenews.org/ 20040828/mathtrek.asp).