# 157

This number is a prime.

The most commonly identified Shiga toxin-producing *Escherichia coli* in North America is *E. coli* O157.

(157^{157} + 1)/(157 + 1) is prime. [Wagstaff]

The smallest prime number for which phi(2*n* + 1) is less than phi(2*n*). [Klee]

The start of the smallest string of consecutive equidigital numbers of length seven. [Santos and Pinch]

The largest odd integer that cannot be expressed as the sum of four distinct nonzero squares with greatest common divisor 1.

The smallest prime of the form 2^{p} + *p*^{3}, where *p* is prime. [Brown]

Two to the power 157 is the smallest "apocalyptic number," i.e., a number of the form 2^{n} that contains '666'. [Pickover]

Samuel Yates, who did an extensive work on the topic of prime period lengths, lived at 157 Capri-D Kings Point in Delray Beach, Florida. He began the list of "Largest Known Primes" (which evolved into *The Prime Pages*).

157 is the smallest three-digit prime that produces five other primes by changing only its first digit: 257, 457, 557, 757, and 857. [Opao]

The smallest prime of form n*((n+1)^(n+2))-(n+3). [Vago]

157 is the smallest irregular prime having an irregularity index of two. [Noe]

Bus number 157 is hijacked in the cop thriller *Dirty Harry*. [May]

The smallest emirp whose binary representation
(10011101_{2}) also forms an emirp in base ten. [Beedassy]

The smallest emirp among sums of distinct emirps. [Capelle]

157^{2} and (157 + 1)^{2} use the same digits.

The smallest balanced emirp. [Punches]

The fourth (and smallest prime) Koechel number of Mozart's string quartets. The fourth largest (i.e., preantepenultimate) such number (KV 499) is again (the largest such) prime. [Hasler]

The smallest prime Köchel number of Mozart's string quartets (String Quartet No. 4 in C). [Hasler]

The smallest emirp whose sum of the digits is another emirp. [Silva]

The smallest emirp number p such that reversal(p^3) is prime. [Firoozbakht]

The product 13 * 17 * 31 * 37 * ... * 149 * 157 = 2847632021876895870149 of consecutive emirps is pandigital. [Poo Sung]

5^{} is close to 157. [Wesolowski]

157 is the smaller prime of non-palindromic dual emirp pairs differing by 10: (157,167);(751,761). Note that reversal(157) =751 and reversal(167)=761. [Hardisky]

The smallest bisexual emirp, i.e., sharing two pairs of sexy primes: (151, 157) and (157, 163). [Loungrides]

The smallest concave emirp. [Loungrides]

Smallest emirp whose binary conversion (10011101 and its reversal 10111001) remain emirps when read in decimal. [Post]

157 is found in the inverse fine-structure constant formula: 157 - 337p/7 = 137.035 999 168, having three prime numbers with the prime constant p; within the uncertainty of the latest determination 137.035 999 173 (35) by Aoyama, T. et al Physical Review Letters, 109, 111807 (2012) arXiv:1205.5368v2. Reference: Sherbon, M.A. "Wolfgang Pauli and the Fine-Structure Constant," Journal of Science, Vol. 2, No. 3, pp.148-154 (2012). [Sherbon]

The smallest emirp with an emirp as its prime index, i.e., π(157)=37. [Gaydos]

The largest rating on the Saffir-Simpson Hurricane Wind Scale occurs at sustained winds of 157 mph or higher.

The 157-foot tower at Cape May has guided ships along the coast and into Delaware Bay since the year Darwin wrote "Origin of Species." [Roxanne]

The Australian rocket frog can cover 157 inches in one leap. [DK]

157 = 4396/28 is a zeroless pandigital expression. [Gaydos]

157 is normally the total number of points for "tricks" in the Swiss card game known as Jass. [Perrenoud]

If we use the English alphabet code a = 1, b = 2, c = 3, … , z = 26, then número primo = 157.

There are 157 7s in the 1401 digits of the first discovered illegal prime (this number includes 161 9s, 157 7s, 151 4s, 146 8s, 141 0s, 135 6s, 131 1s, 130 2s, 125 5s and 124 3s). [Olry]