Are All Perfect Numbers Even?/Progress/Minimum Size/Historical Note
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Historical Note on Minimum Size of Odd Perfect Number
Bryant Tuckerman published a proof in $1968$ that an odd perfect number is greater than $10^{36}$.
Peter Hagis, Jr. published a proof in $1973$ that an odd perfect number is greater than $10^{50}$.
Richard P. Brent and Graeme L. Cohen published a proof in $1989$ that an odd perfect number is greater than $10^{160}$.
Richard P. Brent, Graeme L. Cohen and Hermanus Johannes Joseph te Riele published a proof in $1991$ that an odd perfect number is greater than $10^{300}$.
Pascal Ochem and Michaël Rao published a proof in $2012$ that an odd perfect number is greater than $10^{1500}$.
Sources
- 1968: B. Tuckerman: Odd Perfect Numbers: A Search Procedure, and a New Lower Bound of $10^{36}$ (Not. Amer. Math. Soc. Vol. 15: p. 226)
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Conjecture $2$
- 1973: Peter Hagis, Jr.: A Lower Bound for the Set of Odd Perfect Numbers (Math. Comp. Vol. 27: pp. 951 – 953) www.jstor.org/stable/2005530
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $28$
- 1989: Richard P. Brent and Graeme L. Cohen: A New Bound for Odd Perfect Numbers (Math. Comp. Vol. 53: pp. 431 – 437) www.jstor.org/stable/2008375
- 1989: Richard P. Brent and Graeme L. Cohen: Supplement to A New Lower Bound for Odd Perfect Numbers (Math. Comp. Vol. 53: pp. S7 – S24) www.jstor.org/stable/2008385
- 1991: R.P. Brent, G.L. Cohen and H.J.J. te Riele: Improved Techniques for Lower Bounds for Odd Perfect Numbers (Math. Comp. Vol. 57: pp. 857 – 868) www.jstor.org/stable/2938723
- 2012: Pascal Ochem and Michaël Rao: Odd Perfect Numbers Are Greater than $10^{1500}$ (Math. Comp. Vol. 81: pp. 1869 – 1877) www.jstor.org/stable/23268069