# 37 (number)

**37 (thirty-seven)** is the natural number following 36 and preceding 38. It is an odd prime, the twelfth prime number, and it is best known for a cluster of base-ten arithmetic properties: it divides 111, so every three-digit repdigit is a multiple of it; its reciprocal has a three-digit repeating decimal; and it is the first irregular prime in the theory of [Fermat's Last Theorem](https://www.edgepedia.org/fermats-last-theorem). In popular culture it is the number people most often give when asked for a "random" number between 1 and 100.

| Key fact | Detail |
|---|---|
| Prime index | 37 is the 12th prime number<sup>[1](https://mathworld.wolfram.com/37.html)</sup> |
| Figurate forms | Fourth hex (centered hexagonal) number, \( 37 = 1 + 6 + 12 + 18 \), and third star number, \( 37 = 6 \cdot 3 \cdot 2 + 1 \)<sup>[1](https://mathworld.wolfram.com/37.html)</sup> |
| Repunit factor | \( 111 = 3 \cdot 37 \), so \( 3 \cdot 37 = 111 \), \( 6 \cdot 37 = 222 \), and \( 9 \cdot 37 = 333 \)<sup>[1](https://mathworld.wolfram.com/37.html)</sup> |
| Decimal period | \( 1/37 = 0.027\overline{027} \), period 3; 37 is a unique prime, the only prime with period three<sup>[2](https://mathworld.wolfram.com/UniquePrime.html)</sup> |
| Irregular prime | 37 is the first irregular prime: it divides the numerator of the Bernoulli number \( B_{32} \)<sup>[3](https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat%27s_last_theorem/)</sup> |
| Waring's problem | 37 is the maximum number of fifth powers needed to sum to any number<sup>[4](https://oeis.org/wiki/37)</sup> |
| Chemistry | Atomic number of rubidium (Rb)<sup>[5](https://periodic-table.rsc.org/element/37/Rubidium)</sup> |
| Physiology | Average normal human body temperature, generally accepted as 98.6°F (37°C)<sup>[6](https://medlineplus.gov/ency/article/001982.htm)</sup> |

## Arithmetic properties

**Prime and figurate.** As the 12th prime, 37 sits between 31 and 41 in the prime sequence<sup>[7](https://oeis.org/A000040)</sup>. It is not a twin prime: its neighbors 35 and 39 factor as \( 5 \times 7 \) and \( 3 \times 13 \)<sup>[8](https://www.divinityatlas.com/numbers/37)</sup>. Geometrically it appears in two figurate families. The centered hexagonal numbers, \( 3n(n+1) + 1 \), run 1, 7, 19, 37, 61, and 37 counts the dots in a centered hexagonal figure with four dots on each side<sup>[9](https://oeis.org/A003215)</sup>. The star numbers, \( 6n(n-1) + 1 \), run 1, 13, 37, 73, so 37 is the third star number and its mirror 73 the fourth<sup>[10](https://oeis.org/A003154)</sup>.

**The repunit connection.** The defining decimal property is \( 111 = 3 \cdot 37 \). Because 999 = 27 × 37, we have \( 10^{3} \equiv 1 \pmod{37} \), and this single congruence generates most of the number's tricks. Every three-digit repdigit, 222, 333, up to 999, is divisible by 37 because 111 is<sup>[8](https://www.divinityatlas.com/numbers/37)</sup>. Any three-digit multiple of 37 remains divisible by 37 when its digits are rotated<sup>[11](https://www.johndcook.com/blog/2023/02/12/rotating-multiples-of-37/)</sup>. The divisibility test mirrors the rule of three and nine: split the digits into groups of three from the right and add the groups; the sum is a multiple of 37 if and only if the original number is<sup>[12](https://proofwiki.org/wiki/Divisibility_by_37)</sup>. The reciprocal has the shortest repeating decimal consistent with this: \( 1/37 = 0.\overline{027} \), a period of three digits, which makes 37 a unique prime, a prime whose reciprocal's period is shared with no other prime<sup>[2](https://mathworld.wolfram.com/UniquePrime.html)</sup>.

**Waring's problem.** In the classical problem of representing integers as sums of like powers, 37 is the maximum number of fifth powers needed to sum to any number<sup>[4](https://oeis.org/wiki/37)</sup>.

## The first irregular prime

In 1850 Ernst Eduard Kummer, the German mathematician whose ideal theory reshaped algebra, split the odd primes into regular and irregular classes. A prime \( p \) is regular if it divides none of the numerators of the [Bernoulli numbers](https://www.edgepedia.org/bernoulli-numbers) \( B_2, B_4, \ldots, B_{p-3} \), and Kummer proved Fermat's Last Theorem for every regular exponent<sup>[3](https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat%27s_last_theorem/)</sup>. All primes up to 37 are regular, but 37 itself is not: it divides the numerator of \( B_{32} \)<sup>[3](https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat%27s_last_theorem/)</sup>. The only irregular primes below 100 are 37, 59, and 67, forming the irregular pairs (37, 32), (59, 44), and (67, 58)<sup>[13](https://www.bernoulli.org/)</sup>. Kummer's methods still covered these exponents, proving the theorem beyond exponent 100, and in 1915 Jensen proved that irregular primes, unlike regular ones, are infinite in number<sup>[3](https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat%27s_last_theorem/)</sup>.

## The secretary problem and the 37 percent rule

The number also enters probability through optimal stopping. In the secretary problem, \( n \) candidates are interviewed one at a time in random order, each must be accepted or rejected on the spot, and rejected candidates cannot be recalled. The optimal strategy is to reject the first \( k \) candidates and then hire the first one better than all seen so far, with \( k \sim n/e \)<sup>[14](https://arxiv.org/abs/1512.02996)</sup>. Asymptotically this means rejecting about \( 100/e = 36.7879\ldots \) percent of the candidates, and the probability of hiring the best candidate is \( 1/e = 0.367879\ldots \)<sup>[1](https://mathworld.wolfram.com/37.html)</sup>. The "thirty-seven percent rule" is the popular name for this answer<sup>[14](https://arxiv.org/abs/1512.02996)</sup>.

## Human randomness

**The most "random" number.** When people are asked to pick a number at random, they cluster. In a 2024 Veritasium survey of roughly 200,000 responses to a YouTube community post asking for a random number between 1 and 100, the standout numbers, after setting aside the range ends 1 and 100 and the cultural joke numbers 42 and 69, were 7, 37, 73, and 77; the least-chosen number was 90<sup>[15](https://balabs.bacreativelabs.com/en/deney/bir-sayi-sec)</sup>. The Jargon File, the dictionary of programmer slang, records 37 as the most commonly chosen number in polls for a "random number between 1 and 100"<sup>[16](https://www.catb.org/~esr/jargon/html/R/random-numbers.html)</sup>.

The usual explanation is that people avoid what looks deliberate: even numbers, multiples of 10, repeated digits, and range ends all feel too orderly, leaving numbers built from odd, non-round digits such as 3 and 7<sup>[15](https://balabs.bacreativelabs.com/en/deney/bir-sayi-sec)</sup>. MathWorld notes that the prominence of 37 in such requests is a psychological and cultural phenomenon, not a consequence of its arithmetic properties<sup>[1](https://mathworld.wolfram.com/37.html)</sup>. Magicians exploit the same predictability, steering audiences to 37 with constraints such as two different odd digits below 50<sup>[15](https://balabs.bacreativelabs.com/en/deney/bir-sayi-sec)</sup>.

## 37 in science and computing

**Chemistry.** Rubidium, atomic number 37, is a soft alkali metal discovered in 1861 by [Robert Bunsen](https://www.edgepedia.org/robert-bunsen) and [Gustav Kirchhoff](https://www.edgepedia.org/gustav-kirchhoff), named for the two intense ruby red lines in its spectrum<sup>[5](https://periodic-table.rsc.org/element/37/Rubidium)</sup>. Its \( ^{87}\mathrm{Rb} \) isotope has a resonance frequency near 6.83 GHz and drives the lowest-priced atomic clocks, the small rubidium oscillators used in telecommunications and carried on GPS satellites<sup>[17](https://www.nist.gov/atomic-clocks/how-atomic-clocks-work/clocks-galore)</sup>.

**Physiology.** The average normal body temperature is generally accepted as 98.6°F (37°C), though normal varies by person, age, activity, and time of day, with studies giving a range from 97°F (36.1°C) to 99°F (37.2°C)<sup>[6](https://medlineplus.gov/ency/article/001982.htm)</sup>. The figure dates to 1871, when a German physician derived it from millions of readings; a 2020 Stanford study reported that normal human body temperature has dropped since then<sup>[18](https://time.com/5764051/normal-body-temperature-has-changed/)</sup>.

**Computing.** In the 7-bit ASCII character set, decimal code 37 is the percent sign, %<sup>[19](https://www.ascii-code.com/37)</sup>.

## Numerology and gematria

**Reduction to one.** In digit-reduction numerology, 37 reduces to 1: \( 3 + 7 = 10 \), then \( 1 + 0 = 1 \), so the system says of 37 whatever it says of 1, associated with the Sun, leadership, and new beginnings<sup>[8](https://www.divinityatlas.com/numbers/37)</sup>. A tradition traced to Agrippa in the 16th century calls 37 an "irregular number" requiring special attention<sup>[20](https://numerologywiki.ru/en/number/37)</sup>.

**Genesis 1:1.** The number's best-known mystical connection is arithmetic. The standard Hebrew gematria of [Genesis 1:1](https://www.edgepedia.org/genesis-1-1), the letter values of "In the beginning God created the heavens and the earth," totals 2,701, which factors exactly as \( 37 \times 73 \), a pair of mirror-image primes that are also consecutive star numbers<sup>[21](https://www.mannafest.faith/featured/thirty-seven-and-seventy-three)</sup>. The repunit property also links 37 to 666, the number of the beast, since \( 666 = 6 \times 111 = 18 \times 37 \)<sup>[8](https://www.divinityatlas.com/numbers/37)</sup>. Such relationships are real and verifiable, yet they stem from how numbers are written in base ten, not from anything the ancient writers meant to convey; apart from this arithmetic, 37 lacks any developed lore of its own.<sup>[8](https://www.divinityatlas.com/numbers/37)</sup>

## References

1. [Weisstein, Eric W. "37." Wolfram MathWorld](https://mathworld.wolfram.com/37.html)
2. ["Unique Prime." Wolfram MathWorld](https://mathworld.wolfram.com/UniquePrime.html)
3. ["Fermat's Last Theorem." MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/HistTopics/Fermat%27s_last_theorem/)
4. ["37." OeisWiki](https://oeis.org/wiki/37)
5. ["Rubidium." Royal Society of Chemistry Periodic Table](https://periodic-table.rsc.org/element/37/Rubidium)
6. ["Body temperature norms." MedlinePlus Medical Encyclopedia](https://medlineplus.gov/ency/article/001982.htm)
7. [A000040, The prime numbers. OEIS](https://oeis.org/A000040)
8. ["37." Divinity Atlas, Numbers](https://www.divinityatlas.com/numbers/37)
9. [A003215, Hex (or centered hexagonal) numbers. OEIS](https://oeis.org/A003215)
10. [A003154, Star numbers. OEIS](https://oeis.org/A003154)
11. [Cook, John. "Rotating multiples of 37" (2023)](https://www.johndcook.com/blog/2023/02/12/rotating-multiples-of-37/)
12. ["Divisibility by 37." ProofWiki](https://proofwiki.org/wiki/Divisibility_by_37)
13. ["Irregular pairs." The Bernoulli Number Page](https://www.bernoulli.org/)
14. ["The 'Thirty-seven Percent Rule' and the Secretary Problem with Relative Ranks." arXiv:1512.02996](https://arxiv.org/abs/1512.02996)
15. ["Pick a Number from 1 to 100: Which One Do People Choose?" balabs](https://balabs.bacreativelabs.com/en/deney/bir-sayi-sec)
16. ["random numbers." The Jargon File](https://www.catb.org/~esr/jargon/html/R/random-numbers.html)
17. ["Clocks Galore." NIST](https://www.nist.gov/atomic-clocks/how-atomic-clocks-work/clocks-galore)
18. ["'Normal' Human Body Temperature Has Changed in the Last Century." TIME (2020)](https://time.com/5764051/normal-body-temperature-has-changed/)
19. ["ASCII Code 37, Per cent sign." ascii-code.com](https://www.ascii-code.com/37)
20. ["Number 37 in Numerology: Meaning and Significance." NumerologyWiki](https://numerologywiki.ru/en/number/37)
21. ["Thirty-Seven and Seventy-Three." MannaFest](https://www.mannafest.faith/featured/thirty-seven-and-seventy-three)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Named individual integers*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026*

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