Pervushin's number is \(2,305,843,009,213,693,951 = 2^{61}-1\), the ninth Mersenne prime. It is traditionally denoted as \(M(61)\).

This number was first proven to be prime by Ivan Mikheevich Pervushin in November of 1883 (hence the designation of Pervushin's number).[1] At the time of its discovery, it was the second largest known prime, holding that position until 1911.

The number 61 is one of only nine known numbers holding all three conditions of the New Mersenne conjecture.


Notation Lower bound Upper bound
Scientific notation \(2.305\times10^{18}\) \(2.306\times10^{18}\)
Arrow notation \(12\uparrow17\) \(2\uparrow61\)
Steinhaus-Moser Notation 15[3] 16[3]
Copy notation 2[19] 3[19]
Taro's multivariable Ackermann function A(3,58) A(3,59)
Pound-Star Notation #*(0,3,2,2)*4 #*(6,3,1)*7
BEAF {12,17} {2,61}
Hyper-E notation 2E18 E[2]61
Bashicu matrix system (0)(0)[38967] (0)(0)[38968]
Hyperfactorial array notation 19! 20!
Fast-growing hierarchy \(f_2(55)\) \(f_2(56)\)
Hardy hierarchy \(H_{\omega^2}(55)\) \(H_{\omega^2}(56)\)
Slow-growing hierarchy \(g_{\omega^{\omega+3}3+\omega^{\omega+2}6}(13)\) \(g_{\omega^{\omega2+4}2}(8)\)