# 127

This number is a prime. The smallest odd prime number that cannot be expressed as the sum of a power of two and a prime (so it is also the smallest prime that becomes composite upon changing any one of the bits in its binary expansion). [Capelle] Inge Lehmann would have been exactly 127-years-old when a Google Doodle was dedicated to her. Place a one on both ends of 127 eights to form a prime. [Kulsha] π(127) = 1!3 + 2!3 + 7!3, where n!3 is the triple factorial function. 127 divides (3^3^2 + 2^2^3). [Kulsha] 20 + 21 + 22 + 23 + 24 + 25 + 26 = 127. 127 can be expressed as the sum of factorials of the first three odd numbers (1! + 3! + 5!). John Barrymore kissed Mary Astor and Estelle Taylor a total of 127 times in the film Don Juan (1926). [Dobb] "And Sarah was an hundred and seven and twenty years old: these were the years of the life of Sarah." (Genesis 23:1, KJV) There are 127 prime pairs that sum to ten thousand. [Richstein] 2126 + 7 is the smallest 127-bit prime. [Kulsha] The last three digits of the eleventh Mersenne prime is 127. Note that the next Mersenne prime happens to be M(127). [Honaker] ASCII characters are numbered from 0 to 127. 127 * 1031000 + 1 is prime. [Brown] 127 = 43 * 2 - 1. [Kulsha] (2^127-1)/(2-1) and (2^127+1)/(2+1) are primes. [Luhn] 127 = 5! + 7. Note that 5 and 7 are twin primes. [Avrutin] 127 = 102 + 33. [Trigg] The thickness of a human hair is ~ 1/127 inch. [Gribbin] 27 - 1 is the number of ways of drawing any number of nonintersecting chords among 7 points on a circle. [Rupinski] Mozart, a prime composer, was born on 1/27. [Patterson] In the 19th century, Camille Armand Jules Marie, better known as the "Prince de Polignac," missed the fact that 127 is not an odd number that is the sum of a power of two and a prime. Andy Edwards coined the name "obstinate numbers" to describe such integers. 127 = -1 + 27. (All digits are used in the same order.) [Jeursen] 127 = 1*prime(1) + 2*prime(2) + 7*prime(7). Note that 127 is the largest number with this property. There are only two such numbers, and both are primes. [Firoozbakht] 127 millimeters is exactly five inches. [Vrba] 127 is the first odd prime Motzkin number. [Post] The number of prime-numbered days of the month in a leap year. [Cabisco] The smallest wasteful prime number, which may be described as using fewer letters than in its standard name: "five cubed plus two" rather than "one hundred and twenty-seven." [Beedassy] The smallest prime number that is the concatenation of two distinct cubes (1 and 27). [Cuenta] M(127) = M(M(7)) = M(M(M(3))) = M(M(M(M(2)))). Note that M(2), M(3), M(7) and M(127) are all Mersenne primes. [Capelle] Maya Mohsin Ahmed of UC Davis found that the numbers in an 8-by-8 Franklin square can be described by 127 equations. When Benjamin Franklin (1706-1790) wasn't flying kites, the noted polymath found time to experiment with recreational mathematics. Among other things, he invented magic square variants that are constructed with nonnegative numbers and contain the following properties: the entries of each row and column add to a common (or magic) sum; half of each row or column sums to half of the magic sum; the four corner entries together with the four middle entries add to the magic sum; in addition, each of the "bent rows" (as Franklin called them) have the magic sum. We still do not know what method Franklin used to construct his squares, and leave it for the reader to find other interesting properties. 127 is the 27th prime after 7. [Silva] 127 is the sum of the first (1*2+7) odd primes. [Beedassy] 127 Band is a successful Iranian music group. Their music can best be described as gypsy jazz folk, with an alternative sound. 127 = 10^2+3^3 = gives 127 = 2^2*5^2+3^3 where p(1) appears three times, p(2) twice and p(3) once. 127 = 2^6+3^3+6^2 (palindromic) = p(31) = p(p(11)) = p(p(p(5))) = p(p(p(p(3)))) = p(p(p(p(p(2))))) = p(p(p(p(p(p(1)))))). 127 is the prime for which ((p(1) + ... + p(n))/p(n)) decreases for the first time. [Marot] The largest known prime p such that p2 mod q is odd, where q is the previous prime. [Capelle] The largest known prime number that cannot be written as a sum of cubes greater than one. It's the only such prime written with two cubes concatenated. [Capelle] Smallest odd prime to count the number of ways to Goldbach partition of 10^n as the sum of two primes, when order doesn't matter. [Post] The number of tennis matches required to determine the singles champion at Wimbledon. The smallest regular prime nontrivially a partial sum of regular primes: 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 127, the 25th regular prime. [Post] 127 = prime(1*2*7) + (1*2*7) * π(1*2*7), and is the only known number with this property. [Firoozbakht] I-127 is the only stable isotope of Iodine. [Homewood] The product of the first 127 triangular numbers plus one is prime. [Schiffman] The number of dots in the next term of the following prime sequence puzzle:

```
o o o o o
o o o o         o o o o o o
o o o       o o o o o       o o o o o o o
o o     o o o o     o o o o o o     o o o o o o o o
o o o , o o o o o , o o o o o o o , o o o o o o o o o , ... Why 127?
o o     o o o o     o o o o o o     o o o o o o o o
o o o       o o o o o       o o o o o o o
o o o o         o o o o o o
o o o o o
``` The number of hours that Aron Ralston remained trapped prior to cutting off his own arm with the equivalent of a sharp stick and a dull knife. [Case] The largest sum of digits of the square of a distinct digit brilliant number (924107563^2 = 853974787993798969 and 948302567^2 = 899277758578789489). [Gaydos] According to a statement issued by Abu Dhabi police, 127 cars were involved in the pile-up that left one person of Asian nationality dead and 61 others injured (2 April 2011). [Alwajih] 127 is the maximum number of touchdowns in a season a player can throw for in Madden NFL 06. [Sajak] The smallest of only two primes less than a googol of form 5^x+2, where x is a prime (case x=3). The largest such prime less than a googol is for x=17. [Loungrides] The smallest Friedman prime. [Loungrides] There are 127 species of cuckoos, one of them the infamous road runner. [Homewood] King Achashverosh of the story of Esther ruled 127 provinces. The fifth cuban prime of the first type. [Deacon] The smallest prime that is the arithmetic average of four consecutive primes, i.e., 113, 127, 131, 137. [Loungrides] 127 is the 15th Fibonacci 7-step (heptanacci) number. Note that 2^7-1 = 127, making it the 4th Mersenne prime. [Loungrides] If A = 1, B = 4, C = 1, ..., Z = 8, using the decimal digits of π, then 'A LEFT AND RIGHT TRUNCATABLE PRIME' is a left and right truncatable prime. [Homewood] 127 is found in a "Prime curios" Widget example. The smallest prime de Polygnac number. [Loungrides] The sum of reversals of the first five squares (1+4+9+61+52) as well as the sum of the reversals of the first four cubes (1+8+72+46). [Gaydos] 127 is a triple Mersenne prime, being 2^(2^(2^2-1)-1)-1. [Jones] 2^127 plus its reverse is a prime number. [Gaydos] There are 127 possible prime U.S. Zip Codes that contain exactly two different digits (from 00002 to 99991). [Gaydos] When sunlight hits the moon's surface, the temperature can reach 127 degrees Celsius. [Gupta] The largest prime fact investigated by Dick Sietz in his article "Prime List of Primes" that appeared in the Virginia Mathematics Teacher (Vol. 30, No. 2, 2004). If A=1, B=2, C=3, ..., Z=26, then 'PRIME CURIO' is prime. [Homewood] There are 127 distinct-digit primes for which subtracting the prime index produces an anagram of the prime. [Gaydos] There are 127 prime numbers between 2000 and 3000. [Homewood] Cactaceae comprises 127 genera. [Elsevier] Sonnet 127 of Shakespeare's sonnets is the first of the Dark Lady sequence. [Pierce] The smallest orderly Friedman number. [Olry] Plot the prime points (2,3), (5,7), (11,13), ... , (59,61) to form an irregular n-gon by connecting a straight line to each consecutive points and drawing a line from (2,3) to (59,61) with none of the previous lines being intersected by this line. The area formed is 127 square units. [Bergot] The smallest non-palindromic prime that has the same Collatz trajectory length (46) as its reversal (721). [Gaydos] The 127th prime number (709) and the 127th happy number (907) are an emirp pair. [Gaydos] There are 127 prime days in a leap year. [Homewood] The product of all primes with distinct prime digits plus 127 is prime. [Honaker] The sum of the numbers of all known prime knots on n crossings having Arf invariant 1, i.e., 1 + 1 + 1 + 2 + 4 + 11 + 24 + 83 = 127. [Sariyar] 127 = π(712). [Sariyar] 127 Republicans voted to undo Arizona's election after the Capitol riot. [Baird] Chuck Yeager flew a total of 127 Vietnam War combat missions. A score of 127 in "The Is this prime? game" was achieved by Ravi Fernando, a mathematics graduate student at the University of California, Berkeley, who posted the result in July 2020. The total number of households in Rives, Tennessee (the address of one of the authors), as of the 2000 census.

(There are 7 curios for this number that have not yet been approved by an editor.)