--The

arithmetic progression calculated according to the physical skills of students from the fifth grade was 17.08 repetitions, exceeding the minimum scale by 2.08 repetitions established by SNE; the

arithmetic progression (19.31) calculated according to the performances of the sixth grade exceeds the minimum scale by 3.31 repetitions, and that of the seventh grade (19.89) by 2.89 repetitions.

that is, the sequence [([T.sup.*n][T.sup.n]).sub.n [greater than or equal to] 0] is an

arithmetic progression of strict order m-1 in B(H).

To concentrate on the

arithmetic progressions in the a[b.sub.0]ve formulas Euler removed repeating l's by elementary transformations:

for k = 1, 2, 3, are all

arithmetic progressions with common difference 2k.

Therefore, the new sequence {[M.sub.(k,t]} generated form {[a.sub.n]} is an

arithmetic progression with [M.sub.(0,t)] = t [A.sub.1] = t(t - 1) So.

is a finite

arithmetic progression with common diference [d.sub.1],

We assume without proof the following property of all

arithmetic progressions: the sequence [a.sub.1], [a.sub.2], [a.sub.3], ...

Note that ([2.sup.2m+1], [2.sup.m] + 1) = 1, so from Dirichlet's Theorem we can easily deduce that there are infinitely many primes in the

arithmetic progression:

In fact it is also conjectured that for every n, there are n consecutive primes in

arithmetic progression; at the time of writing, the longest such string consists of 10 primes (see Caldwell, 2004b).

They announced that they had found seven consecutive primes in

arithmetic progression. The previous record had been six.

(i) [for all]k = 0, 1, 2,..., [a.sub.kt+1], [a.sub.kt+2], [a.sub.kt+3],..., [a.sub.kt+t] is a finite

arithmetic progression with [d.sub.1] as the common difference, where t is a constant natural numbers;