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61 (number)
Natural number

61 (sixty-one) is the natural number following 60 and preceding 62.

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In mathematics

61 is the 18th prime number, and a twin prime with 59. As a centered square number, it is the sum of two consecutive squares, 5 2 + 6 2 {\displaystyle 5^{2}+6^{2}} .1 It is also a centered decagonal number,2 and a centered hexagonal number.3

61 is the fourth cuban prime of the form p = x 3 − y 3 x − y {\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}} where x = y + 1 {\displaystyle x=y+1} ,4 and the fourth Pillai prime since 8 ! + 1 {\displaystyle 8!+1} is divisible by 61, but 61 is not one more than a multiple of 8.5 It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ...6

61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number.

61 is the smallest proper prime, a prime p {\displaystyle p} which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length p − 1 , {\displaystyle p-1,} where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, p − 1 10 {\displaystyle {\tfrac {p-1}{10}}} times).7: 166 

In the list of Fortunate numbers, 61 occurs thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number8 (namely 6,469,693,291; 7,420,738,134,871; and 1,922,760,350,154,212,639,131).

There are sixty-one 3-uniform tilings.

Sixty-one is the exponent of the ninth Mersenne prime, M 61 = 2 61 − 1 = 2 , 305 , 843 , 009 , 213 , 693 , 951 {\displaystyle M_{61}=2^{61}-1=2,305,843,009,213,693,951} 9 and the next candidate exponent for a potential fifth double Mersenne prime: M M 61 = 2 2305843009213693951 − 1 ≈ 1.695 × 10 694127911065419641 . {\displaystyle M_{M_{61}}=2^{2305843009213693951}-1\approx 1.695\times 10^{694127911065419641}.} 10

61 is also the largest prime factor in Descartes number,11

3 2 × 7 2 × 11 2 × 13 2 × 19 2 × 61 = 198585576189. {\displaystyle 3^{2}\times 7^{2}\times 11^{2}\times 13^{2}\times 19^{2}\times 61=198585576189.}

This number would be the only known odd perfect number if one of its composite factors (22021 = 192 × 61) were prime.12

61 is the largest prime number (less than the largest supersingular prime, 71) that does not divide the order of any sporadic group (including any of the pariahs).

The exotic sphere S 61 {\displaystyle S^{61}} is the last odd-dimensional sphere to contain a unique smooth structure; S 1 {\displaystyle S^{1}} , S 3 {\displaystyle S^{3}} and S 5 {\displaystyle S^{5}} are the only other such spheres.1314

Notelist

  • R. Crandall and C. Pomerance (2005). Prime Numbers: A Computational Perspective. Springer, NY, 2005, p. 79.
Look up sixty-one in Wiktionary, the free dictionary. Wikimedia Commons has media related to 61 (number).

References

  1. Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers: a(n) is 2*n*(n+1)+1. Sums of two consecutive squares. Also, consider all Pythagorean triples (X, Y, Z equal to Y+1) ordered by increasing Z; then sequence gives Z values.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-09. /wiki/Neil_Sloane ↩

  2. "Sloane's A062786 : Centered 10-gonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A062786 ↩

  3. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A003215 ↩

  4. "Sloane's A002407 : Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A002407 ↩

  5. "Sloane's A063980 : Pillai primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A063980 ↩

  6. "Sloane's A007629 : Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A007629 ↩

  7. Dickson, L. E., History of the Theory of Numbers, Volume 1, Chelsea Publishing Co., 1952. ↩

  8. "Sloane's A005235 : Fortunate numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A005235 ↩

  9. "Sloane's A000043 : Mersenne exponents". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. https://oeis.org/A000043 ↩

  10. "Mersenne Primes: History, Theorems and Lists". PrimePages. Retrieved 2023-10-22. https://t5k.org/mersenne/index.html#unknown ↩

  11. Holdener, Judy; Rachfal, Emily (2019). "Perfect and Deficient Perfect Numbers". The American Mathematical Monthly. 126 (6). Mathematical Association of America: 541–546. doi:10.1080/00029890.2019.1584515. MR 3956311. S2CID 191161070. Zbl 1477.11012 – via Taylor & Francis. https://www.tandfonline.com/doi/full/10.1080/00029890.2019.1584515 ↩

  12. Sloane, N. J. A. (ed.). "Sequence A222262 (Divisors of Descarte's 198585576189.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-27. /wiki/Neil_Sloane ↩

  13. Wang, Guozhen; Xu, Zhouli (2017). "The triviality of the 61-stem in the stable homotopy groups of spheres". Annals of Mathematics. 186 (2): 501–580. arXiv:1601.02184. doi:10.4007/annals.2017.186.2.3. MR 3702672. S2CID 119147703. /wiki/Annals_of_Mathematics ↩

  14. Sloane, N. J. A. (ed.). "Sequence A001676 (Number of h-cobordism classes of smooth homotopy n-spheres.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-10-22. /wiki/Neil_Sloane ↩