--the best values concerning arithmetic progression calculated according to the physical skills of students from the fifth grade was 8.82 repetitions, exceeding the minimum scale by 2.82 repetitions established by SNE; the arithmetic progression (10.18) calculated according to the performances of the sixth grade exceeds the minimum scale by 2.18 repetitions, and that of the seventh grade (11.81) by 1.81 repetitions;

--the arithmetic progression calculated according to the physical skills of students from the fifth grade was 21.06 seconds, exceeding the minimum scale by 1.14 seconds established by SNE; the arithmetic progression (20.15 sec) calculated according to the performances of the sixth grade exceeds the minimum scale by 1.25 seconds, and that of the seventh grade (19.56 sec) by 0.84 seconds.

By Theorem 3 we obtain that T + Q is a (2n - 1)isometry; that is, [([(T + Q).sup.*k][(T + Q).sup.k]).sub.k [greater than or equal to] 0] is an arithmetic progression of order less than or equal to 2n-2.

In the next section we collect some results about applications of arithmetic progressions to m-isometric operators.

Preliminaries: Arithmetic Progressions and (m,g)-Isometries

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.

In that situation both sequences [A.sub.1], [a.sub.2], [a.sub.3], [A.sub.4], [a.sub.5], [a.sub.6] and [b.sub.1], [b.sub.2], [b.sub.3], [b.sub.4], [b.sub.5], [b.sub.6] were arithmetic progressions. That meant that there existed constants c and k such that [a.sub.m+1] = [a.sub.m] + c and [b.sub.m+1] = [b.sub.m] + k for m = 1, 2, ...

Mathematicians have conjectured (but not yet proved) that in the infinite universe of whole numbers, there is no limit to the number of consecutive primes in

arithmetic progression.

A similar argument holds if we consider the

arithmetic progression T(k) - 1 + kT(k)t.

The mathematicians then deduced that the prime numbers are arranged within the spread of almost-primes with enough regularity to ensure that the overall sequence of primes does indeed contain arithmetic progressions of every length.

Green and Tao are now trying to pin down the location of prime arithmetic progressions more precisely.