Beach, James W.||Miller, Herbert (Professor of mathematics)
M.S. Ed. (Master of Education)
Department of Mathematics
After discovering that a pattern existed in the sums of the digits of the squares of integers symbolized in base ten, the writer proceeded to construct and study tables of the sums of digits of the powers of any integer in any given base. It became apparent that both horizontal and vertical patterns evolved, and that for each table a rectangular array of digits repeated. It was the purpose of the study, then, to determine the reasons for the patterns. The writer found that within any given power column of integers expressed by the sums of the digits—or congruent mod (b - 1) where b is the base—a cycle of a maximum of (b - 1) figures repeated. The justification for this pattern is found in the facts that a series of integers in base b can be expressed in terras of the whole numbers 0, 1, 2, . . . , (b-2) mod (b - 1); and that if two integers are congruent mod m, their equivalent powers are likewise congruent. Mathematically, if x ? y mod m, then x^s ? y^s mod m. From the tables, the writer perceived another pattern: the results of any given integer raised to consecutive powers showed a repetition. The writer learned that this cycle was dependent upon Euler's phi-function, designated as ?(m). The digits representing the sums of the digits of the powers of any given integer also repeated. The maximum period of the entire columns was ?(m), where m was the modulus, or m = (b - 1). The basis for this horizontal pattern is a theorem derived by the writer: x^(n+r?(m)) ? x? mod m. Exceptions to this theorem occurred in some cases which are discussed in the paper, but the repetitions began eventually and the theorem became valid at that point. Combining these two theorems, the writer derived the general theorem governing the patterns in all the tables: If x ? y mod m, then x^(n+r?(m)) ? y? mod m.
Anderson, Nancy Ruth, "A study of the sums of the digits of powers of integers in any given base" (1966). Graduate Research Theses & Dissertations. 780.
Northern Illinois University
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