This page contains numbers appearing in number theory.

## Contents

## List of numbers appearing in number theory

**2**is the only even prime number.**6**is the smallest perfect number.**28**is the second perfect number.**60**is the second unitary perfect number.**70**is the smallest weird number.**90**is the third unitary perfect number.**101**is the largest known prime in the form 10^{n}+1.**110**appears in the 290 theorem.**132**is the 6th Catalan number and a pronic number.- It is also is the largest natural number
*n*, such that π^{n}is smaller than the first noncanonical -illion. - 132 is an even number, that has 12 divisors (1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132).
- 132 is digit-reassembly number.
^{[1]}It means that it is equal to the sum of all 2-digits numbers that can be made from the number itself (where the digits can't repeat in each 2-digits number): 12 + 13 + 21 + 23 + 31 + 32 = 132).^{[2]}It's also the smallest number with this property. - Its prime factorization is 2
^{2}× 3 × 11.^{[3]}

- It is also is the largest natural number
**145**appears in the 290 theorem.**163**is the largest Heegner number.**165**is the larger of two known odd unitary superperfect numbers.- The Lucas–Lehmer primality test, which is used for finding the largest known primes, gives
**194**after two iterations. **203**appears in the 290 theorem.**209**is the first composite Kummer number.**210**is the product of the single-digit prime numbers.**231**is the 15th partition number and the 21st triangular number.- Since 21 is the 6th triangular number, 6 the 3rd triangular number, and 3 the 2nd triangular number, it also appears in the calculation of the triangrol, the triangoogol and the triangrolplex.
- It is also equal to 11!!!!.
- Furthermore, it is the number of cubic inches in a U.S. gallon.

**272**is a constructible number and a pronic number.- It is also the largest number
*n*, such that 13^{n}is smaller than a centillion.

- It is also the largest number
**290**is the largest number in the 290 theorem, which is named for it.**341**is the smallest Fermat pseudoprime to base 2.**353**is the smallest number, whose fourth power is the sum of four smaller fourth powers.^{[4]}- Some years in the Hebrew calendar have 353 days.

**385**is the 18th partition number and the 10th square pyramidal number.- Some years in the Hebrew calendar have 385 days.

**462**is the fifth largest known squarefree number of the form_{2n−1}C_{n}.**496**(four hundred ninety-six) is the third perfect number. Its divisors are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.**561**is the first Carmichael number.- 561 is an interesting number primarily for the reason that 561
^{2}-561^{1}-561^{0}=314,159. The resulting number 314159 contains the first six decimal digits of pi. - 561 has 3, 11, and 17 as its prime factors, incidentally the sum of those prime factors is 31, which happens to be the first two decimal digits of pi.

- 561 is an interesting number primarily for the reason that 561
**714**and**715**are a Ruth–Aaron pair.^{[5]}**777**is the 124th lucky number (in the mathematical sense).**1,848**is the largest known idoneal number.**2,047**is the smallest composite Mersenne number with prime index, in this case, (2^{11}−1). The next Mersenne number however, which is 2^{13}−1 or 8,191, is prime.- It is also the smallest strong pseudoprime to base 2.
- In the fast-growing hierarchy, it is equal to
*f*_{2}(8)−1 and*f*_{3}(2)−1.

- The number
**5,040**is the largest known factorial number which is the predecessor of a square number: 7! = 5,040 = 5,041−1 = 71^{2}−1. - The number
**5,041**is equal to 71^{2}. It is the largest known square number which is the successor of a factorial number. **5,775**and**5,776**are the smallest pair of consecutive abundant numbers.**5,777**is the smallest odd composite Stern number.**5,778**is the largest number that is both a triangle and a Lucas number.**5,993**is the largest known odd composite Stern number.**8,128**(eight thousand one hundred twenty-eight) is the fourth perfect number.- The
**8,848**th partition number is the largest one to be smaller than a googol.^{[6]}- It is also the height of the Mount Everest in metres.

**24,310**is the fourth largest known squarefree number of the form_{2n−1}C_{n}.**65,537**is the largest known Fermat prime.^{[7]}**87,360**is the fourth unitary perfect number.**92,378**is the third largest known squarefree number of the form_{2n−1}C_{n}.**1,352,078**is the second largest known squarefree number of the form_{2n−1}C_{n}.- The triple 2 +
**6,436,341**=**6,436,343**is the abc triple with the highest known quality. **33,550,336**(thirty-three millions five hundred fifty thousands three hundred thirty-six) is the fifth perfect number.- Yitang Zhang has proven that there are infinitely many prime gaps not larger than
**70,000,000**.- It is also the prize for correctly answering all sixteen questions in the Indian game show
*Kaun Banega Crorepati*in Indian rupees.

- It is also the prize for correctly answering all sixteen questions in the Indian game show
**1,766,319,049**was shown to be the least*x*for which 61*x*^{2}+1=*y*^{2}for some*y*. This was shown by the Indian mathematician Chāskara in around 1200 AD.^{[8]}**5,425,069,447**and**5,425,069,448**are the smallest pair of consecutive Achilles numbers.**8,589,869,056**is the sixth perfect number. Furthermore, it is the largest known perfect number not containing digit '4'.**262,537,412,640,768,744**is an integer equal to 640,320^{3}+ 744. It is almost equal to the Ramanujan constant.- Its prime factorization is 2
^{3}× 3 × 10,939,058,860,032,031.

- Its prime factorization is 2
**221,256,270,138,418,389,602**is the largest known squarefree number of the form_{2n−1}C_{n}.- It is also equal to 72!/36!
^{2}/2. - Its prime factorization is: 2 · 7 · 13 · 19 · 23 · 37 · 41 · 43 · 47 · 53 · 59 · 61 · 67 · 71, where · denotes multiplication.

- It is also equal to 72!/36!
- A number
*k*≈4.63*10^{103}is given such that for*n*<*k*, there is only one such*n*where "\(\sigma_{od}(n)=\sigma_{od}(n+1)\) and neither*n*nor*n*+1 are perfect squares" holds, namely*n*=1^{[9]}.- The full value of
*k*is**46,305,156,912,921,105,124,676,500,756,345,112,056,691,727,724,000,577,129,664,401,793,869,058,047,789,742,202,704,478,227,034,841,638,012**

- The full value of

## Figurate number collisions

This list contains numbers which are two types of Figurate numbers at the same time.

**36**is the 8th triangular number and the 6th square number.**111**is a Hogben number, a magic constant and a nonagonal number.**120**is the eighth tetrahedral number and the 15th triangular number.**133**is both a Hogben number and an octagonal number.- It is also an idoneal number.
- Furthermore, it is the first number
*n*, such that π^{n}is larger than the first noncanonical -illion. - It’s a product of two different odd primes (7 and 19), so it’s an odd squarefree semiprime.
^{[10]} - And the exceptional Lie algebra E
_{7}has dimension 133.

**136**is a centered triangular number, a centered nonagonal number and the 16th triangular number.- Therefore, it is the number of tiles in a double-15 domino set.
- It is also a constructible number.
- Since it is also approximately equal to the reciprocal of the fine structure constant, it was used in the definition of the Eddington number, which is equal to 136 × 2
^{256}. - Furthermore, it is also the number of floors (namely, concourse, ground and 1 to 138; 41, 74, 110 and 137 have been skipped) served by the main service elevator in the Burj Khalifa.

**175**is both a decagonal number and a magic constant.**176**is a cake number, an octagonal number and a pentagonal number.**210**is the largest number that is both a triangle and a pentatope number.- It is also a pronic number.

**232**is a cake number, a central polygonal number and a decagonal number.- It is also the last number
*n*, such that 20^{n}is smaller than a centillion. - And the isotope thorium-
**232**is the longest-lived actinide nuclide.

- It is also the last number
**252**is both a hexagonal pyramidal number and the number of pips in a double-seven domino set.**286**is both a heptagonal number and a tetrahedral number.- It is also the smallest number
*n*, for which 12^{n}cannot be represented in the double-precision floating-point format. - There are exactly
**286**primordial nuclides. - And there is also a microprocessor with this number in the name.

- It is also the smallest number
**378**is a cake number, a hexagonal number and a triangular number.**560**is both an octagonal number and a tetrahedral number.**576**is both a cake number and a square number.**697**is both a cake number and a heptagonal number.- It is also the last number
*n*, such that e^{n}is smaller than a centillion.

- It is also the last number
**870**is both a magic constant and a pronic number.**946**is both a hexagonal pyramidal number and the number of tiles in a hypothetical double-42 domino set.**969**is both a nonagonal number and a tetrahedral number.**1,105**is a centered square number, a decagonal number and a magic constant.**1,156**is both an octahedral number and a square number.**1,160**is both a cake number and an octagonal number.**1,331**is both a cube number and a centered heptagonal number.**1,379**is both a magic constant and a central polygonal number.**1,540**is the second largest number that is both a triangle and a tetrahedral number.- It is also a decagonal number.

**1,716**is both a house number and a 6-simplex number.**1,771**is both a central polygonal number and a tetrahedral number.**1,794**is both a cake number and a nonagonal number.**2,465**is both a magic constant and an octagonal number.**2,626**is both a cake number and a decagonal number.**2,925**is both a magic constant and a tetrahedral number.**3,003**is the only known number larger than 1 which appears more than six times in Pascal's triangle.**4,900**is the largest number that is both a square and a pyramid number.**4,960**is both a centered triangular number and a tetrahedral number.**7,140**is the largest number that is both a triangle and a tetrahedral number.- It is also the area of a football pitch in square metres.

**10,660**is both a decagonal number and a tetrahedral number.**11,628**is the largest number that is both a triangle and a 5-simplex number.**14,911**is both a hex number and a magic constant.**15,226**is both a cake number and a central polygonal number.**19,600**is the largest number that is both a square and a tetrahedral number.**19,669**is both a heptagonal number and a magic constant.**24,310**is the largest number that is both a triangle and an 8-simplex number.**47,972**is both a cake number and a pentagonal number.**48,620**is both a 9-simplex and a pronic number.**53,130**is both a 5-simplex and a pronic number.**67,600**is both a cake number and a square number.**208,335**is the largest number that is both a triangle and a pyramid number.**226,981**is both a cube number and a star number.**234,136**is both a nonagonal number and a tetrahedral number.**428,536**is both a centered triangular number and a tetrahedral number.**1,004,914**is both a cake number and a centered triangular number.**1,175,056**is both a house number and a square number.**1,414,910**is both a pronic number and a tetrahedral number.**1,582,421**is both a centered square number and a house number.**2,001,000**is both a hexagonal pyramidal number and the number of tiles in a hypothetical double-1999 domino set.**2,287,231**is both a centered pentagonal number and a magic constant.**2,635,752**is both a cake number and a pronic number.**7,759,830**is a hexagonal number, a house number and a triangular number.**9,653,449**is the largest number that is both a square and a stella octangula number.**17,862,376**is both a cake number and a pentagonal number.**90,525,801,730**is both a 31,265-agonal number and a 31,265-agonal pyramidal number.^{[11]}

The full version of Archimedes' cattle problem is partially related to figurate numbers as well.

**20,337,240**is both an octagonal number and a tetrahedral number.**75,203,584**is both a cake number and a square number.**319,118,031**is both a house number and a nonagonal number.

This list contains numbers related to Waring's problem.

**138**is the number of known nonnegative integers which cannot be written as a sum of six nonnegative cubes; the largest of which is 8,042.**223**is the only nonnegative integer which cannot be written as a sum of 36 nonnegative fifth powers.- It is also the largest integer which cannot be written as a sum of 32, 33, 34 or 35 nonnegative fifth powers; there are only fifteen, ten, six and three nonnegative integers with this property, respectively.
- In some countries, such as China, the Band III ends at
**223**MHz. - It is also the number of non-control 8-bit characters.

**239**is the largest integer which cannot be written as a sum of eight nonnegative cubes; the only other nonnegative integer with this property is 23.- It is also one of only seven nonnegative integers which cannot be written as a sum of eighteen fourth powers; the largest integer with this property is 559.

**241**is the number of known nonnegative integers which cannot be written as a sum of four tetrahedral numbers; the largest of which is 343,867.**454**is the largest integer which cannot be written as a sum of seven nonnegative cubes; there are only 17 nonnegative integers with this property.- It is also approximately the number of grams in a pound avoirdupois.
- Furthermore, it is (not considering myriad, lakh and -illiard) the number of numbers with an accepted English name not containing the letter “o”.

**466**is the largest integer which cannot be written as a sum of 28, 29, 30 or 31 nonnegative fifth powers; there are only 52, 41, 31 and 22 nonnegative integers with this property, respectively.**559**is the largest integer which cannot be written as a sum of eighteen fourth powers; there are only seven nonnegative integers with this property.**952**is the largest integer which cannot be written as a sum of 27 nonnegative fifth powers; there are only 66 nonnegative integers with this property.**1,248**is the largest integer which cannot be written as a sum of seventeen fourth powers; there are only 31 nonnegative integers with this property.**4,060**is the number of known nonnegative integers which cannot be written as a sum of five nonnegative cubes; the largest of which is 1,290,740.**8,042**is the largest known integer which cannot be written as a sum of six nonnegative cubes; there are only 138 known nonnegative integers with this property.**13,792**is the largest integer which cannot be written as a sum of sixteen fourth powers; there are only 96 nonnegative integers with this property.**343,867**is the largest known integer which cannot be written as a sum of four tetrahedral numbers; there are only 241 known nonnegative integers with this property.**1,290,740**is the largest known integer which cannot be written as a sum of five nonnegative cubes; there are only 4,060 known nonnegative integers with this property.**113,936,676**is the number of known nonnegative integers which cannot be written as a sum of four nonnegative cubes; the largest of which is 7,373,170,279,850.- Its factorization is 2
^{2}· 3 · 7 · 1,356,389.

- Its factorization is 2
**7,373,170,279,850**is the largest known integer which cannot be written as a sum of four nonnegative cubes; there are only 113,936,676 known nonnegative integers with this property.- Its prime factorization is 2 × 5
^{2}× 18,521 × 7,961,957.

- Its prime factorization is 2 × 5

## Approximations of these numbers

For 8,042:

Notation | Approximation |
---|---|

Scientific notation | \(8.042\times 10^3\) (exact) |

Arrow notation | \(2\uparrow {13}\) |

BEAF | \(\{2,13\}\) |

Fast-growing hierarchy | \(f_2(9)<n<f_2(10)\) |

For 221,256,270,138,418,389,602:

Notation | Approximation |
---|---|

Scientific notation | \(2.2126 \times 10^{20}\) |

Arrow notation | \(136↑10\) |

BEAF | \(\{136,10\}\) |

Chained arrow notation | \(136→10\) |

Fast-growing hierarchy | \(f_2(65)\) |

Hardy hierarchy | \(H_{\omega^2}(65)\) |

Slow-growing hierarchy | \(g_{\omega^{10}}(136)\) |

## See also

## Sources

- ↑ https://en.wikipedia.org/wiki/Digit-reassembly_number
- ↑ http://oeis.org/search?q=132%2C264%2C396&language=english&go=Search
- ↑ https://www.wolframalpha.com/input/?i=132
- ↑ Euler's sum of powers conjecture
- ↑ Notable Properties of Specific Numbers (MROB) - 714
- ↑ OEIS A000041
- ↑ OEIS, Sequence A019434. Accessed 2021-05-28.
- ↑ C. Selenius, [Rationale of the Chakravāla Process https://pdf.sciencedirectassets.com/272588/1-s2.0-S0315086000X00894/1-s2.0-0315086075901433/main.pdf] (p.179). Accessed 2021-05-28.
- ↑ KimIkikardesLiMa, On the problem σ
_{od}(n)=σ_{od}(n+1) (2010) (p.557) - ↑ https://www.wolframalpha.com/input/?i=133
- ↑ Numberphile, 90,525,801,730 Cannon Balls (accessed 2021-01-23)

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