70 (number)

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

Mathematics

Properties of the integer

70 is the fourth discrete sphenic number, as the first of the form <math>2 \times 5 \times r</math>.<ref>Template:Cite web</ref> It is the smallest weird number, a natural number that is abundant but not semiperfect,<ref>Template:Cite web</ref> 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.<ref>Template:Cite web</ref>Template:Efn It is also the sixth Pell number, preceding the tenth prime number 29, in the sequence <math>\{0, 1, 2, 5, 12, 29, \ldots\}</math>.

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

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.<ref>Template:Cite OEIS</ref>

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,<ref name="A002808">Template:Cite OEIS</ref> 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),<ref name="A002808" /> the third non-trivial member of the 43-aliquot tree.

The sum 43 + 50 + 40 = 133 represents the one-hundredth composite number,<ref name="A002808" /> 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).<ref>Template:Cite OEIS</ref><ref name="A006450" />Template:Efn

Figurate numbers

• 70 is the seventh pentagonal number.<ref>Template:Cite web</ref>
• 70 is also the fourth 13-gonal (tridecagonal) number.<ref>Template:Cite web</ref>
• 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 <math>\{1, 2, 6, 20\}</math>, 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.<ref>Template:Cite OEIS</ref>

Geometric properties

7-simplex

Two-dimensional orthographic projection of the 7-simplex, a uniform 7-polytope with seventy tetrahedral cells

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:

<math>\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}</math>

Aside from the 7-simplex, there are a total of seventy other uniform 7-polytopes with <math>\mathrm {A_7}</math> 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.<ref>Template:Cite web</ref>

Leech lattice

The sum of the first 24 squares starting from 1 is 70Template:Sup = 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 religion

• In Jewish tradition, 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 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 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, Template:Langx; Template:Langx, short for Template:Langx. (For French, this is true only in France, Canada and Luxembourg; other French-speaking regions such as Belgium, Switzerland, Aosta Valley and Jersey use Template:Lang.)<ref>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".</ref>