--the

arithmetic progression calculated according to the physical skills of students from the fifth grade was 18.33 repetitions, exceeding the minimum scale by 2.33 repetitions established by SNE; the

arithmetic progression (19.35) calculated according to the performances of the sixth grade exceeds the minimum scale by 2.35 repetitions, and that of the seventh grade (20.11) by 2.11 repetitions.

By (5), the operator sequence [([T.sup.*n][T.sup.n]).sub.n [greater than or equal to] 0] is an

arithmetic progression of strict order m-1.

Note that, in each interval, the terms are in

arithmetic progression (AP) with common diference [d.sub.1].

In the event that the 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] are both

arithmetic progressions (1), one can readily show that the sum of the six elements chosen as described above is

Mathematicians and others had identified numerous examples involving six consecutive primes in

arithmetic progression, but no examples of seven.

We call the finite

arithmetic progression [a.sub.kt+1], [a.sub.kt+2], [a.sub.kt+3],..., [a.sub.kt+t] is the (k + 1)th periodic of {[a.sub.n]} and [a.sub.(k+1)t], [a.sub.(k+1)t+1] is the (k + 1)th interval of {[a.sub.n]}, [d.sub.1] is named the common difference inside the periods and [d.sub.2] is called the interval common difference, t is called the number sequence {[a.sub.n]}'s period.

We shall make use of Dirichlet's Theorem on primes in

arithmetic progression in the following form.

(Readers do, however, need some familiarity with number theory at an undergraduate level and should have taken a first course in modern algebra.) The author begins with a thorough introduction to elementary prime number theory, including Dirichlet's theorem on primes in

arithmetic progressions, the Brun sieve, and the Erdos-Selberg proof of the prime number theorem.

Montgomery, Primes in

arithmetic progressions, Mich.

Just in case you thought that magic squares were just amusing diversions and had no relevance to any other areas of mathematics, this very problem (construction of a 3 by 3 magic square all of whose entries are squares) has been shown to be highly relevant to problems in various domains, including

arithmetic progressions, Pythagorean triangles, and elliptic curves.