# 61

This number is a prime.

It is not known whether the double-Mersenne number *MM*(61)
is prime or composite. It contains 694127911065419642
digits and is currently far too large for the usual
Lucas-Lehmer test.

The smallest prime whose reversal is a square. [Trigg]

61 divides 67 * 71 + 1. *Honaker's problem* asks if there are three larger consecutive primes *p* < *q* < *r* such that *p*|*q**r* + 1.

One way to convert words to numbers is to let A = 1, B = 2, C = 3, ... , Z = 26; and then compute the sum. Using this "alphabet code," the word PRIME is prime.

Roger Maris ended the season with 61 home runs in '61.

Bob Dylan wrote a song (and album) called "Highway 61 Revisited."

You can arrange 61 coins into a hexagonal pattern with one coin in the center.

61°S Latitude 61°W Longitude is in the Drake Passage. [Faidley]

The number of digits in *M*(61) is 61 turned upside down.

The average gestation period of a dog is 61 days.

In 1657, Fermat challenged the mathematicians of Europe and England, "We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul (Fermat's region) will." Among the challenges was this 500-year-old example from Bhaskara II: *x*^{2} - 61*y*^{2} = 1 (*x*, *y* > 0). [Balazs]

(2^{61} - 1)/(2 - 1) and (2^{61} + 1)/(2 + 1) are both primes. [Luhn]

Louis A. Bloomfield's textbook *How Things Work* examines 61 objects in
our everyday world.

The letter 'A' occurs 61 times among the U.S. state names. It is the most common letter used in this manner. [Blanchette]

61 is the only known example where the reversal of a Mersenne prime exponent gives a power of 2. [Luhn]

The 61st book in the KJV Bible has 61 verses. [Desrosiers]

61 = 115 in base 1 + 1 + 5. [Poo Sung]

61 is the (6 - 1)th repfigit number: 61, 6 + 1 = 7, 1 + 7 = 8, 7 + 8 = 15, 8 + 15 = 23, 15 + 23 = 38, 23 + 38 = 61. [Mirizzi]

Ten consecutive distinct digits begin at position 61 after the decimal point of π. [Wu]

(61+1)^61-61 is prime. [Luhn]

61! - 60! + 59! - 58! + ... + 5! - 4! + 3! - 2! + 1! is prime.

The smallest multidigit prime such that 61*16 + 1 is prime. [Gupta]

The smallest multidigit prime p such that the sum of first 2p primes is prime. [Russo]

The smallest multidigit prime p such that the sum of digits of p^{p} is a square. [Russo]

The smallest prime that can be written as the sum of a prime number of primes to prime powers in a prime number of ways: 2^{2} + 2^{3} + 7^{2}, 2^{2} + 5^{2} + 2^{5}, and 3^{2} + 5^{2} + 3^{3}. [Hartley]

The least prime such that its square can be expressed as the sum of one, two, three, or four distinct squares:
61^{2} = 60^{2} + 11^{2} = 52^{2} + 24^{2} + 21^{2} = 56^{2} + 22^{2} + 10^{2} + 1^{2}. [Rupinski]

The smallest prime for which the decimal expansion of the reciprocal 1/61 contains all 10 digits in equal proportions. [Hartley]

The largest known prime of the form *Ack*(*m*, *m*), where *Ack* is the Ackermann function (which grows faster than an exponential function). *Ack*(3, 3) = 61. [Hartley]

The average life expectancy for an American male in 1960. Note the same year it was 73 (also prime) for American females. [Fearon]

61 is the sum of two consecutive squares: 5^{2} + 6^{2}. [Schlesinger]

There are 61 abstract groups of order 1464. Note that 61 divides 1464. [Hartley]

61 is the smallest number m and only two-digit number *m*
such that the reversal of concatenation of the first *m*
natural numbers is prime. [Firoozbakht]

Promethium (atomic number 61) may be missing from our solar system, but it has been detected in the spectrum of a variable star in the constellation Andromeda.

(61!/16!)+1 is prime with prime digital sum and length. [Patterson]

The smallest prime p such that p# is pandigital. [Patterson]

Cyclist Lance Armstrong defeated Jan Ullrich by 61 seconds in the 2003 Tour de France. [Hartley]

61 codons specify individual amino acids. [Necula]

61 countries took part in World War II.

61!*16! - 1 is prime with prime sum of digits. [Firoozbakht]

In the tiny hamlet of Pinghan, nestled deep among a stand of limestone hills in a remote region of southwestern China, locals honor an old national tradition of buying a coffin at the age of 61.

61 is the 7th prime of the form 3k+1 and 61 = (77+777)/(7+7). Note that seven 7's are used. [Firoozbakht]

The first of five consecutive primes that can be arranged in a 3-by-3 square whose columns and rows sum to consecutive primes (199, 211, 223):

61 67 71 67 71 73 71 73 79.[Silva]

61 is the sum of the first three primes ending with seven. [Sladcik]

Adding all the Roman numerals in Yugoslavia (i.e., yUgosLaVIa = 5+50+5+1 = 61) is prime. [Opao]

61 is a prime Euler number. [Post]

Figurately speaking, 61 is both the 6th (6*1) centered square number and the 5th (6-1) centered hexagonal number. [Post]

When numbering the sixty vertices of the fullerine buckyball in ordered pairs of ascending and descending numbers, the sum of the opposite pairs will equal 61. [McGown]

The first prime overlooked by Mersenne in his erroneous conjectured list of exponents. [Beedassy]

The smallest prime integer quotient of a 4-digit Harshad number, i.e., 61 = 1098/(1+0+9+8). [Brueggeman]

In 1838, astronomer Friedrich Wilhelm Bessel calculated the large proper motion of 61 Cygni using trigonometric parallax. This was the first star (other than the Sun) to have its distance known from the Earth. [McGown]

61 (sixty-one) is the largest prime whose name spelled out in English has no repeated letters. [Brahinsky]

61 is the smallest base *b* for which the generalized
repunit ((*b*^37)-1)/(*b*-1) is prime.

The Monument, a memorial to the Great Fire of London in 1666, is the world's highest (at 61 meters) stone column marking the distance to Pudding Lane, where the fire started. Note that the reversal of 1666 is a beastly prime ending with 61. [Capelle]

The smallest prime repfigit number equal to the sum of the first n repfigit numbers. [Loungrides]

The smallest prime which can be represented as a sum of emirps (61 = 13 + 17 + 31). [Poo Sung]

The 61st Fibonacci number (i.e., 2504730781961) is the smallest pandigital Fibonacci number. Note that it also ends in 61. [Gupta]

In "The Local Galaxy Group & Galactic Neighborhood Observing Guide" (Astronomical League, 2008) by Robert McGown and Dareth Murray, the authors dedicate the Virgo galaxy M61 (Prime Curio Galaxy) to the research of Chris K. Caldwell, G. L. Honaker, Jr., and others whose prime number research brings together some of the unique mathematical relationships in the universe.

The number of "questions and answers" in a complete TV game show of
*Jeopardy!* [Davidson]

The smallest prime of order n that divides the sum of the first n composite numbers. [Silva]

The first Self-Belgian number of type 2. [Angelini]

No odd Fibonacci number is divisible by the prime 61. [Schiffman]

The great Topologist John Milnor had his conjecture disproved by M. Mahowald, who found S(61), the set of all oriented diffeomorphism classes of closed smooth homotopy 61-spheres is the trivial group. Reference: "Differential Toplogy Forty-Six Years Later", John Milnor, Notice of the AMS, June/July 2011, pp.804-809. [Post]

"And the asses were thirty thousand and five hundred; of
which the LORD'S tribute was **threescore and one**." (*Numbers* 31:39, KJV)

Rotation (or 61) is a pocket billiards game. [Junie]

A leap second on Saturday (6/30/2012) will cause a 61-second minute.

The Focke-Wulf Fw 61 is often considered the first practical, functional helicopter.

Astronomers have discovered a large asteroid containing abundant amounts of water orbiting a white dwarf star, designated GD 61.

The sum of the first n primes type 4k+1 is equal to 2^x, just for two n values, i.e., n=4 and n=8, producing the following sums: 5+13+17+29=64=2^6 and 5+13+17+29+37+41+53+61=256=2^8. Will this ever be satisfied again for another pair of n and x values? [Rivera]

The reversal of 61 is a fourth power, and the reversal of its fourth power is a prime. Are there any more primes with both of these properties? [Gaydos]

The 61 1st-move pairs in chess yielding the minimum 19 2nd-move options for White are the symmetrical double advances of King-Bishop Pawns and all 20 responses with either that pawn's or either Rook Pawn's single advance. [Merickel]

61 is the smallest prime p such that 2p+3 is a perfect cube. Can you find next prime with this property? [Honaker]

Prime numbers that end with "61" occur more often than any other 2-digit ending among all primes less than one billion. [Rivera]

The sum of the lowest digit in each of the powers of two (2 + 4 + 8 + 1 + 2 + 4 + 1 + 2 + 1 + 0 + 0 + 0 + 1 + 1 + 2 + 3 + 0 ...) is 61 unless there are some large power of two that has no zero. [Gaydos]

The 61st decade (601 to 610) was the first to have no semiprime year. [Gaydos]

If we sum the last digits of the numbers up to 61 raised to the power of zerro or bigger, we always yield primes (61, 271 or 199). 61 is the smallest number with this property. If we sum the last digits of the first 61 numbers n raised to the power of n, the result is prime again (283). [Petrov]

"Class of '61" is an American war drama television film produced by Steven Spielberg.

The smallest prime number with only one "6" that converts to a semiprime if the 6 is changed to a 9. The sequence begins 61, 269, 463, 601, etc. See PC10000...14363 (1000-digits). [Jeremy]

The earliest prime to appear in the decimal expansion of golden ratio. [Gupta]

61 and its prime index (18) are both reverse fourth powers. There are no more examples of this property in the first billion primes. Are there any more at all? [Gaydos]

Exactly half (61) of the first 122 emirps are distinct-digit numbers. [Gaydos]