# 167

This number is a prime.

2 * 3^{4} + 5 is prime. [Gundrum]

(296/167)^{2} is closer to π than Ptolomy's 377/120. Note that 167 is prime. [Moody]

In 2003, outgoing Governor George Ryan of Illinois commuted the death sentences of 167 death row inmates two days before leaving office, calling the death penalty process "arbitrary and capricious, and therefore immoral."

There are two 167-digit numbers consisting of only two distinct digits which are of the form p*2^167 with p prime. They correspond to the numbers containing only the digits 6 & 7, and 8 & 9. [Rupinski]

The number of prime quadruples, counting (3, 5, 7, 11), below one million. (A prime quadruple is four consecutive primes, such that the first and the last differ by 8. It has the form (p, p+2, p+6, p+8) for p > 3).

All highly cototient numbers greater than 167 are congruent to 9 (mod 10).

167 = 5*7 + (2^2)*3*11, proving primality. Largest "additive" case using all candidate prime factors in one equation. [Wood]

Wieferich proved that 167 is the only prime requiring exactly eight cubes to express it. [Rupinski]

At the start of backgammon each player has a pip count of 167. [La Haye]

The smallest number whose fourth power begins with four identical digits.

The least prime factor of the smallest vampire number that has prime fangs of the same length (117067 = 167 * 701). A vampire number is an integer which can be written as the product of two factors (its "fangs") whose digits together are a rearrangement of the original number.

The smallest multidigit prime such that the product of digits is equal to the number of digits times the sum of the digits, i.e., 1 * 6 * 7 = 3(1 + 6 + 7).

Bangkok officially has the longest place name on the planet. When transliterated into English, its full Thai name rambles on for 167 letters.

It is conjectured that every positive integer may be
written as the sum of seven nonnegative cubes except these
seventeen: {15, 22, 23, 50, 114, 167, 175, 186, 212, 231,
238, 239, 303, 364, 420, 428, 454}. This conjecture has
been tested to 10^{34} and proven for numbers
divisible by four. [Caldwell]

InuYasha is a Japanese manga series which ran for 167 episodes.

Outlaws Bonnie and Clyde received 167 bullet holes in their stolen Ford Deluxe near Bienville Parish, Louisiana.

A prime number that meets a perfect square in New York City. See picture. [Green]

The smallest emirp p such that reversal(p*reversal(p)) is prime. [Firoozbakht]

191^181 = 179^173 (mod 167). Note that 167, 173, 179, 181 and 191 are five consecutive primes. [Firoozbakht]

The smallest emirp p such that both numbers prime(p) and prime(prime(p)) are emirp. [Firoozbakht]

The smallest emirp derived from a two-digit emirp by inserting its digital difference in between. [Beedassy]

Only about one in 167 people have AB blood type with a negative Rh factor.

Admiral Horatio Nelson landed a force of 167 seamen and marines when he assaulted the French garrison of the Turks Islands.

The smallest emirp whose index is also emirp, i.e. 167 is the 13th emirp. [Loungrides]

The ancient aqueduct at Segovia, Spain, features a total of 167 arches.

The smallest prime that is the concatenation of a square and a prime in two ways {1,67 and 16,7}. [Vantieghem]

The only 3-digit prime p such that the product of first p odd numbers minus 2 is prime, i.e., 1*3*5*…*331*333-2. Note that there are also a single-digit prime (p=3) and a double-digit prime (p=37) with the same property. [Loungrides]

A 1905 Indian Head Penny in certified mint state condition could bring as much as $167 at auction.

Is 167# the largest primorial number that contains a unique digit? It contains the digit 8 only once. [Gaydos]

The first positive integer, more precisely a prime, whose fourth power begins with four identical digits: 167⁴=777796321. [Leonardis]

The 167th prime number (991) and the 167th unhappy number (199) are an emirp pair. 167 is also an emirp. [Gaydos]

The smallest distinct-digit 3-digit emirp whose a digit is equal to the sum of the other two. Note that there are two pairs of such emirps, i.e., (167, 761) and (347, 743). [Loungrides]