127 (number)

← 126 127 128 →
← 120 121 122 123 124 125 126 127 128 129 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred twenty-seven
Ordinal	127th (one hundred twenty-seventh)
Factorization	prime
Prime	31st
Divisors	1, 127
Greek numeral	ΡΚΖ´
Roman numeral	CXXVII, cxxvii
Binary	11111112
Ternary	112013
Senary	3316
Octal	1778
Duodecimal	A712
Hexadecimal	7F16

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

In mathematics

127 as a centered hexagonal number

• 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, ${\displaystyle 2^{127}-1}$, 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.
  • ${\displaystyle 2^{127}-1}$ is the largest prime ever discovered by hand calculations as well as the largest known double Mersenne prime.
  • Furthermore, 127 is equal to ${\displaystyle 2^{7}-1}$, and 7 is equal to ${\displaystyle 2^{3}-1}$, 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 ${\displaystyle p={\frac {x^{3}-y^{3}}{x-y}}}$, ${\displaystyle x=y+1}$. ^[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 ${\displaystyle 127=2^{7}-1\,}$, as well as binary since ${\displaystyle 1111111=(1+1)^{111}-1\,}$ .
• 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: ${\displaystyle 127=3+5+7+11+13+17+19+23+29}$. ^[7]
• 127 is the smallest odd number that cannot be written in the form ${\displaystyle p+2^{x}}$, for p is a prime number, and x is an integer, since ${\displaystyle 127-2^{0}=126,}$${\displaystyle 127-2^{1}=125,}$${\displaystyle 127-2^{2}=123,}$${\displaystyle 127-2^{3}=119,}$${\displaystyle 127-2^{4}=111,}$${\displaystyle 127-2^{5}=95,}$ and ${\displaystyle 127-2^{6}=63}$ are all composite numbers. ^[8]
• 127 is an isolated prime where neither ${\displaystyle p-2}$ nor ${\displaystyle p+2}$ 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 ${\displaystyle 127=1\cdot {\textrm {prime}}(1)+2\cdot {\textrm {prime}}(2)+7\cdot {\textrm {prime}}(7),}$ where ${\displaystyle {\textrm {prime}}(n)}$ is the nth prime number. There are only two numbers with that property; the other one is 43.
• 127 is equal to ${\displaystyle {\textrm {prime}}^{6}(1),}$ where ${\displaystyle {\textrm {prime}}(n)}$ 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.
2. Sloane, N. J. A. (ed.). "SequenceA109611(Chen primes: primes p such that p + 2 is either a prime or a semiprime.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. "Sloane's A051634 : Strong primes". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
4. "Sloane's A003215 : Hex (or centered hexagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
5. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-27.
6. Sloane, N. J. A. (ed.). "SequenceA024916(sum_{k=1..n} sigma(k) where sigma(n) = sum of divisors of n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
7. Sloane, N. J. A. (ed.). "SequenceA071148". 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.
8. Sloane, N. J. A. (ed.). "SequenceA006285(Odd numbers not of form p + 2^x (de Polignac numbers))". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
9. Sloane, N. J. A. (ed.). "SequenceA137985". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.. Complementing any single bit in the binary representation of these primes produces a composite number.
10. Sloane, N. J. A. (ed.). "SequenceA065577(Number of Goldbach partitions of 10^n)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-08-31.

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 136 - 138