← 39 40 41 →
Cardinalforty
Ordinal40th
(fortieth)
Numeral systemquadragesimal
Factorization23 × 5
Divisors1, 2, 4, 5, 8, 10, 20, 40
Greek numeralΜ´
Roman numeralXL
Latin prefixquadrage-
Binary1010002
Ternary11113
Senary1046
Octal508
Duodecimal3412
Hexadecimal2816
ArmenianԽ
Hebrewמ / ם
Babylonian numeral𒐏

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[1][2] and is now the standard form.

Mathematics

Forty is the fourth octagonal number.[3] As the sum of the first four pentagonal numbers: , it is also the fourth pentagonal pyramidal number.[4] Forty is a repdigit in ternary, and a Harshad number in decimal.[5]

40 is an abundant number, because the sum of its proper divisors is greater than 40.

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.[6] 40 is also the ninth refactorable number.[7]

40 is the algebraic polynomial degree of six-cycle logistic map, [8]

Forty is the number of n-queens problem solutions for .[9]

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 twelfth triangular number). 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,

[10]

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.[11]

Given 40, the Mertens function returns , as with 39; the only other smaller number to return a value of zero is 2.[12] Adding 39 and 40 yields the twenty-second indexed prime, 79, equal to the sum of all six lucky numbers of Euler .[10]

40 also lies between the 8th pair of sexy primes (37, 43)[13] — the former the difference of 2 and 39, that also multiply to 78 — which represent the only two points in the set of natural numbers where the ratio of prime numbers to composite numbers (in adjacency, up to) is .[14][15][a]

In science

Astronomy

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 the 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.

Judaism

  1. He went up on the seventh day of Sivan, after God gave the Torah to the Jewish people, in order to learn the Torah from God, and came down on the seventeenth day of Tammuz, when he saw the Jews worshiping the Golden Calf and broke the tablets (Deuteronomy 9:11).
  2. He went up on the eighteenth day of Tammuz to beg forgiveness for the people's sin and came down without God's atonement on the twenty-ninth day of Av (Deuteronomy 9:25).
  3. He went up on the first day of Elul and came down on the tenth day of Tishrei, the first Yom Kippur, with God's atonement (Deuteronomy 10:10).

Christianity

Christianity similarly uses forty to designate important time periods.[16]

Islam

Yazidism

Funerary customs

Hinduism

In the Hindu system some of the popular fasting periods consist 40 days and is called the period One 'Mandala Kalam' Kalam means a period and Mandala Kalam means a period of 40 days. For example, the devotees (male and female) of Swami Ayyappa, the name of a Hindu god very popular in Kerala, India (Sabarimala Swami Ayyappan) strictly observed forty days fasting and visit (since female devotees of a certain biological age group would not be able to perform the continuous 40-day-austerities, they would not enter into the god's temple until September 2018) with their holy submission or offerings on 41st or a convenient day after a minimum 40 days practice of fasting. The offering is called "Kaanikka".

Buddhism

Sikhism

In entertainment

In sports

In other fields

See also: List of highways numbered 40 and List of highways numbered 40A

Forty is also:

See also

Notes

  1. ^ 28 is the only other earlier point, where there are eighteen composite numbers, and nine prime numbers (at 23); also, where 40 is the twenty-seventh composite number, it can also be interpreted as the twenty-eighth non-prime in [15]

References

  1. ^ Google nGrams
  2. ^ Oxford English Dictionary, 1st edition, s.v.
  3. ^ "Sloane's A000567 : Octagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  4. ^ "Sloane's A002411 : Pentagonal pyramidal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  5. ^ "Sloane's A005349 : Niven (or Harshad) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  6. ^ "Sloane's A005835 : Pseudoperfect (or semiperfect) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A033950". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-10.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A118454 (Algebraic degree of the onset of the logistic map n-bifurcation.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-02-29.
  9. ^ Sloane, N. J. A. (ed.). "Sequence A000170 (Number of ways of placing n nonattacking queens on an n X n board.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-10.
  10. ^ a b Sloane, N. J. A. (ed.). "Sequence A014556". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-09.
  11. ^ Euler, Leonhard (1772). Extrait d'un lettre de M. Euler le pere à M. Bernoulli concernant le Mémoire imprimé parmi ceux de 1771 (Extract of a letter). Nouveaux Mémoires de l'Académie Royale des Sciences de Berlin (in French). University of the Pacific (The Euler Archive). pp. 35, 36.
  12. ^ "Sloane's A028442 : Numbers n such that Mertens' function is zero". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A156274 (List of prime pairs of the form (p, p+6).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-11.
  14. ^ Sloane, N. J. A. (ed.). "Sequence A00040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
  15. ^ a b Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x > 1 and y > 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-01-16.
  16. ^ a b Michael David Coogan, A Brief Introduction to the Old Testament: The Hebrew Bible in Its Context, Oxford, 2008, p. 116
  17. ^ "Flogging".
  18. ^ Qur'an 5:25–26
  19. ^ Qur'an 7:142
  20. ^ Sesame Street. Season 40. Episode 4187. November 10, 2009. PBS.
  21. ^ Dallal, Tamalyn (2007). 40 Days & 1001 Nights. Seattle: Melati Press. back cover. ISBN 978-0979515507.
  22. ^ "40 Days & 1001 Nights – One Woman's Dance Through Life in the Islamic World".

Further reading