Highest Power of 2 Dividing Numerator of Sum of Odd Reciprocals/Historical Note
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Historical Note on Highest Power of 2 Dividing Numerator of Sum of Odd Reciprocals
This result was posed as an elementary problem by John Lewis Selfridge in March $1960$: Problems for Solution: E1406-E1410 (American Mathematical Monthly Vol. 67: p. 290) www.jstor.org/stable/2309704.
He notes that H.S. Shapiro and D.L. Slotnick leave the problem unsolved in an article in a $1959$ commercial publication, where they suggest that:
- an estimate [of this] power of $2$ ... seems in general to be a difficult number theoretic problem.
In the event, $7$ contributors are reported as having submitted a solution, of which that by D.L. Silverman was the one published.
Among the solvers was Donald E. Knuth, who included the problem as an exercise of difficulty level $M33$ in his The Art of Computer Programming: Volume 1: Fundamental Algorithms.
Sources
- 1959: H.S. Shapiro and D.L. Slotnick: On the Mathematical Theory of Error-Correcting Codes (IBM J. Res. Develop. Vol. 3: pp. 25 – 34)
- 1960: J. Selfridge and D.L. Silverman: E1408: The Highest Power of $2$ in the Numerator of $\sum_{i \mathop = 1}^k 1 / \paren {2 i - 1}$ (Amer. Math. Monthly Vol. 67: pp. 924 – 925) www.jstor.org/stable/2309478
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.7$: Harmonic Numbers: Exercise $18$