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1. The Thirteenth Century was in but its second year when Fibonacci (or Pisano) proposed a problem on the number of offspring of a pair of rabbits, whose solution led to the sequence of numbers now named after him. There is reason to believe that Fermat derived many of his arithmetic theorems from a knowledge of recurrences and, certainly, his celebrated Last Theorem may be stated as a problem on sequences. Lucast was the first to make any extended researches on sequences, establishing a great many properties of certain second order sequences. Carmichael [1 ]1 in 1920 made the first attack on sequences in general, and established their fundamental property of modular periodicity. This paper undertakes a general survey of the modular properties of linear recurring sequences, beginning from the results of a paper by H. T. Engstrom and two by Morgan Ward. ? No problems on sequences are considered here which are not strictly modular, though questions on divisibility sequencesjj and their remarkable factorization properties are closely related. The mechanism which the author uses for examining the properties of sequences is an isomorphism between the set of all sequences satisfying a fixed recurrence and a polynomial ring of operators. The isomorphism is not with the abstract ring but with a particular realization of it, and this is not especially surprising as a linear sequence is essentially an exponential function. This isomorphism may be derived from the theory of generating functions, and includes the fundamental identity used in Ward [11 ]. In Chapter II the isomorphism is set up and the basic properties of the ring are examined and their interpretation is given for the sequences. For example, the zero divisors of the ring correspond to the sequences which satisfy recurrences of lower order.

An isomorphism between linear recurring sequences and algebraic rings