Misplaced Pages

70 (seventy) is the natural number following 69 and preceding 71.

70 is the value ${\displaystyle n}$ whose factorial is closest to a googol, where ${\displaystyle 70!\approx 1.1978571\ldots \times 10^{100}}$.

Mathematics

Properties of the integer

70 is the fourth discrete sphenic number, as the first of the form ${\displaystyle 2\times 5\times r}$. It is the smallest weird number, a natural number that is abundant but not semiperfect, where it is also the second-smallest primitive abundant number, after 20. 70 is in equivalence with the sum between the smallest number that is the sum of two abundant numbers, and the largest that is not (24, 46).

70 is the tenth Erdős–Woods number, since it is possible to find sequences of seventy consecutive integers such that each inner member shares a factor with either the first or the last member. It is also the sixth Pell number, preceding the tenth prime number 29, in the sequence ${\displaystyle \{0,1,2,5,12,29,\ldots \}}$.

70 is a palindromic number in bases 9 (77₉), 13 (55₁₃) and 34 (22₃₄).

Happy number

70 is the thirteenth happy number in decimal, where 7 is the first such number greater than 1 in base ten: the sum of squares of its digits eventually reduces to 1. For both 7 and 70, there is

${\displaystyle 49\mapsto 16+81\mapsto 97\mapsto 81+49\mapsto 130\mapsto 1+9\mapsto 10\mapsto 1.}$

97, which reduces from the sum of squares of digits of 49, is the only prime after 7 in the successive sums of squares of digits (7, 49, 97, 130, 10) before reducing to 1. More specifically, 97 is also the seventh happy prime in base ten.

70 = 2 × 5 × 7 simplifies to 7 × 10, or the product of the first happy prime in decimal, and the base (10).

Aliquot sequence

70 contains an aliquot sum of 74, in an aliquot sequence of four composite numbers (70, 74, 40, 50, 43) in the prime 43-aliquot tree.

• The composite index of 70 is 50, which is the first non-trivial member of the 43-aliquot tree.
• 40, the Euler totient of 100, is the second non-trivial member of the 43-aliquot tree.
• The composite index of 100 is 74 (the aliquot part of 70), the third non-trivial member of the 43-aliquot tree.

The sum 43 + 50 + 40 = 133 represents the one-hundredth composite number, where the sum of all members in this aliquot sequence up to 70 is the fifty-ninth prime, 277 (this prime index value represents the seventeenth prime number and seventh super-prime, 59).

Figurate numbers

• 70 is the seventh pentagonal number.
• 70 is also the fourth 13-gonal (tridecagonal) number.
• 70 is the fifth pentatope number.

The sum of the first seven prime numbers aside from 7 (i.e., 2, 3, 5, 11, …, 19) is 70; the first four primes in this sequence sum to 21 = 3 × 7, where the sum of the sixth, seventh and eighth indexed primes (in the sequence of prime numbers) 13 + 17 + 19 is the seventh square number, 49.

Central binomial coefficient

70 is the fourth central binomial coefficient, preceding ${\displaystyle \{1,2,6,20\}}$, as the number of ways to choose 4 objects out of 8 if order does not matter; this is in equivalence with the number of possible values of an 8-bit binary number for which half the bits are on, and half are off.

Geometric properties

7-simplex

In seven dimensions, the number of tetrahedral cells in a 7-simplex is 70. This makes 70 the central element in a seven by seven matrix configuration of a 7-simplex in seven-dimensional space:

${\displaystyle {\begin{bmatrix}{\begin{matrix}8&7&21&35&35&21&7\\2&28&6&15&20&15&6\\3&3&56&5&10&10&5\\4&6&4&70&4&6&4\\5&10&10&5&56&3&3\\6&15&20&15&6&28&2\\7&21&35&35&21&7&8\end{matrix}}\end{bmatrix}}}$

Aside from the 7-simplex, there are a total of seventy other uniform 7-polytopes with ${\displaystyle \mathrm {A_{7}} }$ symmetry. The 7-simplex can be constructed as the join of a point and a 6-simplex, whose order is 7!, where the 6-simplex has a total of seventy three-dimensional and two-dimensional elements (there are thirty-five 3-simplex cells, and thirty-five faces that are triangular).

70 is also the fifth pentatope number, as the number of 3-dimensional unit spheres which can be packed into a 4-simplex (or four-dimensional analogue of the regular tetrahedron) of edge-length 5.

Leech lattice

The sum of the first 24 squares starting from 1 is 70 = 4900, i.e. a square pyramidal number. This is the only non trivial solution to the cannonball problem, and relates 70 to the Leech lattice in twenty-four dimensions and thus string theory.

In science

70 is the atomic number of ytterbium, a lanthanide.

In religion

• In Jewish tradition:
  • There is a core of 70 nations and 70 world languages, paralleling the 70 names in the Table of Nations.
  • There were 70 men in the Great Sanhedrin, the Supreme Court of ancient Israel. (Sanhedrin 1:4.)
  • According to the Jewish Aggada, there are 70 perspectives ("faces") to the Torah (Numbers Rabbah 13:15).
  • Seventy elders were assembled by Moses on God's command in the desert (Numbers 11:16–30).
  • Psalm 90:10 allots three score and ten (70 years) for a man's life, and the Mishnah attributes that age to "strength" (Avot 5:32), as one who survives that age is described by the verse as "the strong".
  • Ptolemy II Philadelphus ordered 72 Jewish elders to translate the Torah into Greek; the result was the Septuagint (from the Latin for "seventy"). The Roman numeral seventy, LXX, is the scholarly symbol for the Septuagint.

• In Christianity:
  • In Matthew 18:21–22, Jesus tells Peter to forgive people seventy times seven times.
  • In Luke 10:1–24, Jesus appoints Seventy Disciples and sends them out in pairs to preach the Gospel.
• Seventy is a priesthood office in the Latter Day Saint religion.

• In Islamic history and in Islamic interpretation the number 70 or 72 is most often and generally hyperbole for an infinite amount:
  • There are 70 dead among the Prophet Muhammad's adversaries during the Battle of Badr.
  • 70 of the Prophet Muhammad's followers are martyred at the Battle of Uhud.
  • In Shia Islam, there are 70 martyrs among Imam Hussein's followers during the Battle of Karbala.

In law

In certain cases, copyrights expire after 70 (or 50) years, especially after the death of the latest author (see, Berne Convention).

In other fields

• In some traditions, 70 years of marriage is marked by a platinum wedding anniversary.
• Under Social Security (United States), the age at which a person can receive the maximum retirement benefits (and may do so and continue working without reduction of benefits).

Number name

Several languages, especially ones with vigesimal number systems, do not have a specific word for 70: for example, French: soixante-dix, lit. 'sixty-ten'; Danish: halvfjerds, short for halvfjerdsindstyve, 'three and a half score'. (For French, this is true only in France; other French-speaking regions such as Belgium, Switzerland, Aosta Valley and Jersey use septante.)

Notes

1. The smallest sequence of seventy consecutive integers sharing a factor with either first or last member starts at the twenty-three digit number (with decimal representation), 26214699169906862478864 = 2 × 3 × 7 × 11 × 13 × 19 × 23 × 29 × 37 × 43 × 47 × 53 × 67 × 73 × 2221, or approximately 2.62 × 10. Its largest prime factor is the sixty-seventh super-prime, where 70 lies midway between the thirteenth pair of sexy primes (67, 73).
2. It is also a Harshad number in bases 6, 8, 9, 10, 11, 13, 14, 15 and 16.
3. Meanwhile, the aliquot sum of 164 = 74 + 40 + 50 is 130, with a sum-of-divisors of 294, and an arithmetic mean of divisors of 49.

References

1. "Sloane's A007304 : Sphenic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
2. "Sloane's A006037 : Weird numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
3. "Sloane's A059756 : Erdős-Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
4. Sloane, N. J. A. (ed.). "Sequence A059757 (Initial terms of smallest Erdős-Woods intervals corresponding to the terms of A059756.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
5. ^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
6. Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
7. Sloane, N. J. A. (ed.). "Sequence A007770 (Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
8. Sloane, N. J. A. (ed.). "Sequence A035497 (Happy primes: primes that eventually reach 1 under iteration of "x -> sum of squares of digits of x".)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
9. ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
10. Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
11. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
12. Sloane, N. J. A. (ed.). "Sequence A000203 (...the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
13. Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
14. Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-31.
15. "Sloane's A000326 : Pentagonal numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
16. "Sloane's A051865 : 13-gonal (or tridecagonal) numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
17. Sloane, N. J. A. (ed.). "Sequence A000984 (Central binomial coefficients: binomial(2*n,n) as (2*n)!/(n!)^2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
18. "Sloane's A000332 : Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
19. Peter Higgins, Number Story. London: Copernicus Books (2008): 19. "Belgian French speakers however grew tired of this and introduced the new names septante, octante, nonante etc. for these numbers".

External links

• Integers