40 (forty) is the natural number following 39 and preceding 41.
Though the word is related to four (4), the spelling forty replaced fourty during the 17th century and is now the standard form.
In mathematics
Forty is the fourth octagonal number. As the sum of the first four pentagonal numbers: , it is also the fourth pentagonal pyramidal number. Forty is a repdigit in ternary, and a Harshad number in decimal.
40 is the smallest number with exactly nine solutions to the equation Euler's totient function (for values 41, 55, 75, 82, 88, 100, 110, 132, and 150 of ).
Adding up some subsets of the divisors of 40 (e.g., 1, 4, 5, 10, and 20) gives 40; hence, 40 is the ninth semiperfect number. 40 is also the ninth refactorable number.
Forty is the number of -queens problem solutions for .
Euler's lucky numbers
Swiss mathematician Leonard Euler noted forty prime numbers generated by the quadratic polynomial , with values : 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, and 1601.
The differences between terms are 0, 2, 4, 6, 8, ..., 78 (equivalently, up through a difference of twice 39). The first such prime (41) is the thirteenth prime number, where 13 divides the largest thrice over. The last such prime (1601) is the two hundred and fifty-second prime number (where 252 is the sum between two through twenty-two, inclusive) as well as one more than the square of forty, 402 = 1600. Importantly, 41 is also the largest of six lucky numbers of Euler of the form,
These forty prime numbers are the same prime numbers that are generated using the polynomial with values of from 1 through 40, and are also known in this context as Euler's "lucky" numbers.
Given 40, the Mertens function returns 0, as with 39 — the only other smaller number to return a value of zero is 2. Adding 39 and 40 yields a prime number, the twenty-second indexed prime 79. It is equal to the sum of all six lucky numbers of Euler, specifically . The largest number to have a totient value of 40 is 150, which also returns 0 for the Mertens function.
In science
The atomic number of zirconium. Negative forty is the unique temperature at which the Fahrenheit and Celsius scales correspond; that is, −40 °F = −40 °C. It is referred to as either "minus forty" or "forty below".
Astronomy
Messier object M40, a magnitude 9.0 double star in the constellation Ursa Major
The New General Catalogue object NGC 40, a magnitude 12.4 planetary nebula in the constellation Cepheus
In religion
The number 40 is found in many traditions without any universal explanation for its use. In Jewish, Christian, Islamic, and other Middle Eastern traditions it is taken to represent a large, approximate number, similar to "umpteen".
Sumerian
Enki ( /ˈɛŋki/) or Enkil (Sumerian: dEN.KI(G)𒂗𒆠) is a god in Sumerian mythology, later known as Ea in Akkadian and Babylonian mythology. He was originally patron god of the city of Eridu, but later the influence of his cult spread throughout Mesopotamia and to the Canaanites, Hittites and Hurrians. He was the deity of crafts (gašam); mischief; water, seawater, lake water (a, aba, ab), intelligence (gestú, literally "ear") and creation (Nudimmud: nu, likeness, dim mud, make bear). He was associated with the southern band of constellations called stars of Ea, but also with the constellation AŠ-IKU, the Field (Square of Pegasus). Beginning around the second millennium BCE, he was sometimes referred to in writing by the numeric ideogram for "40", occasionally referred to as his "sacred number".
A large number of myths about Enki have been collected from many sites, stretching from Southern Iraq to the Levantine coast. He figures in the earliest extant cuneiform inscriptions throughout the region and was prominent from the third millennium down to Hellenistic times.
The exact meaning of his name is uncertain: the common translation is "Lord of the Earth": the Sumerian en is translated as a title equivalent to "lord"; it was originally a title given to the High Priest; ki means "earth"; but there are theories that ki in this name has another origin, possibly kig of unknown meaning, or kur meaning "mound". The name Ea is allegedly Hurrian in origin while others claim that it is possibly of Semitic origin and may be a derivation from the West-Semitic root *hyy meaning "life" in this case used for "spring", "running water". In Sumerian E-A means "the house of water", and it has been suggested that this was originally the name for the shrine to the God at Eridu.
