Are All Perfect Numbers Even?/Progress/Minimum Size/Historical Note

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