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127 (number)
Natural number

127 (one hundred [and] twenty-seven) is the natural number following 126 and preceding 128. It is also a prime number.

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In mathematics

  • As a Mersenne prime, 127 is related to the perfect number 8128. 127 is also the largest known Mersenne prime exponent for a Mersenne number, 2 127 − 1 {\displaystyle 2^{127}-1} , which is also a Mersenne prime. It was discovered by Édouard Lucas in 1876 and held the record for the largest known prime for 75 years.
    • 2 127 − 1 {\displaystyle 2^{127}-1} is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime.
    • Furthermore, 127 is equal to 2 7 − 1 {\displaystyle 2^{7}-1} , and 7 is equal to 2 3 − 1 {\displaystyle 2^{3}-1} , and 3 is the smallest Mersenne prime, making 7 the smallest double Mersenne prime and 127 the smallest triple Mersenne prime.
  • There are a total of 127 prime numbers between 2,000 and 3,000.
  • 127 is also a cuban prime of the form p = x 3 − y 3 x − y {\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}} , x = y + 1 {\displaystyle x=y+1} .1 The next prime is 131, with which it comprises a cousin prime. Because the next odd number, 129, is a semiprime, 127 is a Chen prime.2 127 is greater than the arithmetic mean of its two neighboring primes; thus, it is a strong prime.3
  • 127 is a centered hexagonal number.4
  • It is the seventh Motzkin number.5
  • 127 is a palindromic prime in nonary and binary.
  • 127 is the first Friedman prime in decimal. It is also the first nice Friedman number in decimal, since 127 = 2 7 − 1 {\displaystyle 127=2^{7}-1\,} , as well as binary since 1111111 = ( 1 + 1 ) 111 − 1 {\displaystyle 1111111=(1+1)^{111}-1\,} .
  • 127 is the sum of the sums of the divisors of the first twelve positive integers.6
  • 127 is the smallest prime that can be written as the sum of the first two or more odd primes: 127 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 {\displaystyle 127=3+5+7+11+13+17+19+23+29} .7
  • 127 is the smallest odd number that cannot be written in the form p + 2 x {\displaystyle p+2^{x}} , for p is a prime number, and x is an integer, since 127 − 2 0 = 126 , {\displaystyle 127-2^{0}=126,} 127 − 2 1 = 125 , {\displaystyle 127-2^{1}=125,} 127 − 2 2 = 123 , {\displaystyle 127-2^{2}=123,} 127 − 2 3 = 119 , {\displaystyle 127-2^{3}=119,} 127 − 2 4 = 111 , {\displaystyle 127-2^{4}=111,} 127 − 2 5 = 95 , {\displaystyle 127-2^{5}=95,} and 127 − 2 6 = 63 {\displaystyle 127-2^{6}=63} are all composite numbers.8
  • 127 is an isolated prime where neither p − 2 {\displaystyle p-2} nor p + 2 {\displaystyle p+2} is prime.
  • 127 is the smallest digitally delicate prime in binary.9
  • 127 is the 31st prime number and therefore it is the smallest Mersenne prime with a Mersenne prime index.
  • 127 is the largest number with the property 127 = 1 ⋅ prime ( 1 ) + 2 ⋅ prime ( 2 ) + 7 ⋅ prime ( 7 ) , {\displaystyle 127=1\cdot {\textrm {prime}}(1)+2\cdot {\textrm {prime}}(2)+7\cdot {\textrm {prime}}(7),} where prime ( n ) {\displaystyle {\textrm {prime}}(n)} is the nth prime number. There are only two numbers with that property; the other one is 43.
  • 127 is equal to prime 6 ( 1 ) , {\displaystyle {\textrm {prime}}^{6}(1),} where prime ( n ) {\displaystyle {\textrm {prime}}(n)} is the nth prime number.
  • 127 is the number of non-equivalent ways of expressing 10,000 as the sum of two prime numbers.10

In other fields

  • The non-printable "Delete" (DEL) control character in ASCII.
  • Linotype (and Intertype) machines used brass matrices with one of 127 possible combinations punched into the top to enable the matrices to return to their proper channel in the magazine.

References

  1. "Sloane's A002407 : Cuban primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27. https://oeis.org/A002407 ↩

  2. Sloane, N. J. A. (ed.). "Sequence A109611 (Chen primes: primes p such that p + 2 is either a prime or a semiprime.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩

  3. "Sloane's A051634 : Strong primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27. https://oeis.org/A051634 ↩

  4. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27. https://oeis.org/A003215 ↩

  5. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27. https://oeis.org/A001006 ↩

  6. Sloane, N. J. A. (ed.). "Sequence A024916 (sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩

  7. Sloane, N. J. A. (ed.). "Sequence A071148". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.. Partial sums of a sequence of odd primes; a(n) = sum of the first n odd primes. /wiki/Neil_Sloane ↩

  8. Sloane, N. J. A. (ed.). "Sequence A006285 (Odd numbers not of form p + 2^x (de Polignac numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. /wiki/Neil_Sloane ↩

  9. Sloane, N. J. A. (ed.). "Sequence A137985". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.. Complementing any single bit in the binary representation of these primes produces a composite number. /wiki/Neil_Sloane ↩

  10. Sloane, N. J. A. (ed.). "Sequence A065577 (Number of Goldbach partitions of 10^n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-31. /wiki/Neil_Sloane ↩