# 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 $\sum_{i = 1}^k 1 / \left({2 i - 1}\right)$*(*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 $18$