Euler's lucky numbers

Swiss mathematician Leonard Euler noted forty prime numbers generated by the quadratic polynomial ${\displaystyle n^{2}+n+41}$, with values ${\displaystyle n=0,1,2,...,39{\text{ }}}$:

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 ${\displaystyle n}$ 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,

${\displaystyle m^{2}-m+n;{\text{ }}m=\{0,\cdots ,n-1\}.}$^[9]

These forty prime numbers are the same prime numbers that are generated using the polynomial ${\displaystyle n^{2}-n+41}$ with values of ${\displaystyle n}$ from 1 through 40, and are also known in this context as Euler's "lucky" numbers.^[10]

Given 40, the Mertens function ${\displaystyle M(n)}$ returns ${\displaystyle 0}$, as with 39; the only other smaller number to return a value of zero is 2.^[11] Adding 39 and 40 yields the twenty-second indexed prime, 79, equal to the sum of all six lucky numbers of Euler ${\displaystyle \{2,3,5,11,17,41\}}$.^[9]

40 also lies between the 8th pair of sexy primes (37, 43)^[12] — 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 ${\displaystyle \mathbb {N} }$ where the ratio of prime numbers to composite numbers (in adjacency, up to) is ${\displaystyle {\tfrac {1}{2}}}$.^{[13][14][lower-alpha 1]}

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

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

• In the Hebrew Bible, forty is often used for time periods, forty days or forty years, which separate "two distinct epochs".^[15]
• Rain fell for "forty days and forty nights" during the Flood (Genesis 7:4).
• Noah waited for forty days after the tops of mountains were seen after the flood, before releasing a raven (Genesis 8:5–7).
• Spies were sent by Moses to explore the land of Canaan (promised to the children of Israel) for "forty days" (Numbers 13:2, 25).
• The Hebrew people lived in the lands outside of the promised land for "forty years". This period of years represents the time it takes for a new generation to arise (Numbers 32:13).
• Several early Hebrew leaders and kings are said to have ruled for "forty years", that is, a generation. Examples include Eli (1 Samuel 4:18), Saul (Acts 13:21), David (2 Samuel 5:4), and Solomon (1 Kings 11:42).
• Goliath challenged the Israelites twice a day for forty days before David defeated him (1 Samuel 17:16).
• Moses spent three consecutive periods of "forty days and forty nights" on Mount Sinai:

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).

• A mikvah consists of 40 se'ah (approximately 200 U.S. gallons or 760 liters) of water
• The prophet Elijah had to walk 40 days and 40 nights before arriving at mount Horeb (1 Kings 19:8).
• 40 lashes is one of the punishments meted out by the Sanhedrin (Deuteronomy 25:3), though in actual practice only 39 lashes were administered.^[16]
• (Numbers 14:33–34) alludes to the same with ties to the prophecy in The Book of Daniel. "For forty years—one year for each of the forty days you explored the land—you will suffer for your sins and know what it is like to have Me against you."
• One of the prerequisites for a man to study Kabbalah is that he is forty years old.
• "The registering of these men was carried on cruelly, zealously, assiduously, from the rising of the sun to its going down, and was not brought to an end in forty days" (3 Maccabees 4:15).
• Jonah warns Nineveh that "Forty days more, and Nineveh shall be overthrown." (Jonah 3:4)

Christianity

Christianity similarly uses forty to designate important time periods.^[15]

• Before his temptation, Jesus fasted "forty days and forty nights" in the Judean desert (Matthew 4:2, Mark 1:13, Luke 4:2).
• Forty days was the period from the resurrection of Jesus to the ascension of Jesus (Acts 1:3).
• According to Stephen, Moses' life is divided into three 40-year segments, separated by his growing to adulthood, fleeing from Egypt, and his return to lead his people out (Acts 7:23, 30, 36).
• In modern Christian practice, Lent consists of the 40 days preceding Easter. In much of Western Christianity, Sundays are excluded from the count, there are 46 days in total of Lent; in Eastern Christianity, Sundays are included.
• The Forty Martyrs of Sebaste
• Kirk Kilisse, "Forty Churches" (Σαράντα Εκκλησιές) in Eastern Thrace
• Rain fell for "forty days and forty nights" during the Flood (Genesis 7:4).

Islam

• Muhammad was forty years old when he first received the revelation delivered by the archangel Gabriel.
• Masih ad-Dajjal roams around the Earth in forty days, the first day length is like one year, the second day is like one month, the third day is like one week and the next day (until the 40th day) is like one day.
• God forbade the Israelites from entering the Holy Land for 40 years to separate them from Musa (Moses) and his brother.^[17]
• Musa (Moses) spent 40 days on Mount Sinai where he received the 10 commandments.^[18]

Yazidism

• In the Yazidi faith, The Chermera Temple (meaning "40 Men" in the Yazidi dialect) is so old that no one remembers how it came to have that name but it is believed to derive from the burial of 40 men on the mountaintop site.

Funerary customs

• Some Russians, Bulgarians, and Serbs believe that ghosts of the dead linger at the site of their death for forty days. After the forty days, additional prayers are performed at the grave (parastos (парастос) or panihida (панихида)), to escort the soul on its way to God's court.
• Many Christian Filipinos mark the end of the initial mourning period on the fortieth day after death, and have a Mass said. They believe that the soul remains on the earthly plane for forty days before entering the afterlife, recalling how Christ ascended to heaven forty days after his Resurrection.

Hinduism

• In Hinduism, some popular religious prayers consist of forty shlokas or dohas (couplets, stanzas). The most common being the Hanuman Chalisa (chaalis is the Hindi term for 40).

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

• When discussing meditation, the Visuddhimagga suggests 40 meditation subjects to focus on.

Sikhism

• Anand Sahib, the fifth and the final of the daily Sikh prayers have 40 paragraphs, and the 40th paragraph is often read when concluding any Sikh ceremony.
• Chali Mukte (40 liberated ones) refer to the 40 soldiers in the army of Guru Gobind Singh. These 40 disciples were the most favorite and beloved disciples of the Guru.