71 (number)

← 70 71 72 →
← 70 71 72 73 74 75 76 77 78 79 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	seventy-one
Ordinal	71st (seventy-first)
Factorization	prime
Prime	20th
Divisors	1, 71
Greek numeral	ΟΑ´
Roman numeral	LXXI, lxxi
Binary	10001112
Ternary	21223
Senary	1556
Octal	1078
Duodecimal	5B12
Hexadecimal	4716

71 (seventy-one) is the natural number following 70 and preceding 72.

In mathematics

71 is the 20th prime number. Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. ^{[1] [2]}

71 is a centered heptagonal number. ^[3]

It is a regular prime, ^[4] a Ramanujan prime, ^[5] a Higgs prime, ^[6] and a good prime. ^[7]

It is a Pillai prime, since ${\displaystyle 9!+1}$ is divisible by 71, but 71 is not one more than a multiple of 9. ^[8] It is part of the last known pair (71, 7) of Brown numbers, since ${\displaystyle 71^{2}=7!+1}$. ^[9]

71 is the smallest of thirty-one discriminants of imaginary quadratic fields with class number of 7, negated (see also Heegner numbers). ^[10]

71 is the largest number which occurs as a prime factor of an order of a sporadic simple group, the largest (15th) supersingular prime. ^{[11] [12]}

See also

• 71 (disambiguation)

References

1. Sloane, N. J. A. (ed.). "SequenceA006567(Emirps (primes whose reversal is a different prime))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
2. Baker, Alan (January 2017). "Mathematical spandrels". Australasian Journal of Philosophy. 95 (4): 779–793. doi:10.1080/00048402.2016.1262881. S2CID   218623812.
3. Sloane, N. J. A. (ed.). "SequenceA069099(Centered heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
4. "Sloane's A007703 : Regular primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. "Sloane's A104272 : a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
6. "Sloane's A007459 : a(n+1) = smallest prime > a(n) such that a(n+1)-1 divides the product (a(1)...a(n))^2". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. "Sloane's A028388 : prime(n) such that prime(n)^2 > prime(n-i)*prime(n+i) for all 1 <= i <= n-1". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
8. Sloane, N. J. A. (ed.). "SequenceA063980(Pillai primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Berndt, Bruce C.; Galway, William F. (2000). "On the Brocard–Ramanujan Diophantine equation ${\displaystyle n!+1=m^{2}}$". Ramanujan Journal. 4 (1): 41–42. doi:10.1023/A:1009873805276. MR   1754629. S2CID   119711158.
10. Sloane, N. J. A. (ed.). "SequenceA046004(Discriminants of imaginary quadratic fields with class number 7 (negated).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-08-03.
11. Sloane, N. J. A. (ed.). "SequenceA002267(The 15 supersingular primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
12. Duncan, John F. R.; Ono, Ken (2016). "The Jack Daniels problem". Journal of Number Theory. 161: 230–239. doi: 10.1016/j.jnt.2015.06.001 . MR   3435726. S2CID   117748466.