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10 (twelve, dozen, onety) is the base of the numeral system of this wiki.

An interesting thing is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + X + E + 10 + ... = −1/10 = −0.1, i.e.

$ \sum_{k=1}^{\infty}(k)=-\frac{1}{10}=-0.1 $ (this number is

$ \zeta(-1) $ , which equals the negative value of the reciprocal of 10, or the additive inverse of the multiplicative inverse of 10) i.e.

$ \zeta(-1)=-\frac{1}{10}=-0.1 $

The negative value of the reciprocal of 10 (i.e. −1/10) is the value counter-intuitively ascribed to the series 1+2+3....

10 is a composite number, the smallest positive integer with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 10. 10 is also a highly composite number, the next one being 20 (=2×10), and all highly composite numbers ≥10 are divisible by 10.

10 is the smallest abundant number.

10 is the average of a twin prime pair (E and 11), i.e. both 10−1 and 10+1 are primes.

10 is the least common multiple of the n such that anxn+an-1xn-1+...+a1x+a0=0 have algebraic solution: {1, 2, 3, 4}.

10 is the least common multiple of the all-harshad numbers: {1, 2, 4, 6}. (10 itself is a harshad number in all bases except base 8 (octal))

The numbers n such that all primes not dividing n are = ±1 mod n (i.e. either of the form nk+1 or of the form nk−1) are exactly the proper divisors of 10: {1, 2, 3, 4, 6}, and the least common multiple of these numbers are exactly 10.

The numbers n such that cos(2pi/n) is rational (in fact, 2cos(2pi/n) is integer) are exactly the proper divisors of 10: {1, 2, 3, 4, 6}, and the least common multiple of these numbers are exactly 10.

10 is the ratio of 20 and 2, where 20 is the largest n such that lambda(n)=2, and 2 is the largest n such that lambda(n)=1, where lambda is the Carmichael lambda function.

10 is the least common multiple of 1 to 4.

10 is the greatest common divisor of the order of unsolvable groups ≤ 14000.

10 is the greatest common divisor of the composite order of simple groups ≤ 14000.

10 is the product of the n such that there are no Greco-Latin squares of order n: {2, 6}.

The star numbers are exactly the centered n-gonal numbers for n=10.

10 is the value of

$ \frac{\pi^2}{(\ln 2) \cdot (\ln \lambda)} $ , where

$ \lambda $ is the Levy’s constant. 10 is the smallest nonnegative even number n such that the numerator of

$ \frac{\zeta(n)}{\pi^n} $ is not 1.

10 is the smallest nonnegative even number n such that the numerator of Bn has at least one prime factor ≥ n+3, where Bn is the Bernoulli number.

It is known that

$ \zeta(n) $ is irrational for n=3, and for at least one other odd n≤10.

A number n has terminate reciprocal if and only if n is regular to 10, i.e. all prime divisors of n are prime divisors of 10, besides, a number n has reciprocal with only 1 digit after the dozenal point if and only if n is divisor of 10.

The n-th Fibonacci number F(n) has terminate reciprocal if and only if n is divisor of 10.

10 is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 14) is greater than itself.

10 is the smallest positive integer which is not power of squarefree number (i.e. not in OEIS A072774).

10 is the smallest positive integer which is neither squarefree nor prime power.

10 is the smallest positive integer which is neither squarefree nor perfect power.

10 is the smallest positive integer which is neither squarefree nor powerful.

10 is the smallest number n>1 such that the n-th cyclotomic polynomial has no root mod p for all primes p <= n.

10 is the smallest number k such that k/(largest power of squarefree kernel of k) is larger than 1, i.e. 10/6 = 2 > 1.

The highly composite numbers contains the first 5 multiples of 10 (10, 20, 30, 40, and 50), and no other numbers between them.

10 is the only number which is both (equal to mn) and (average of 2m and 2n) for some integer pairs (m, n) such that m and n are consecutive integers (i.e. n = m+1). ((m, n) = (3, 4))

10 is the largest number n such that if 0≤k≤n−1 and gcd(k, n) = 1 (i.e. k and n are coprime), then n+k is prime.

Both of 10−1 and 10+1 are primes, in fact, all of 10±1, (10/2)±1, and (10/3)±1 are primes, the next such number is 220990, besides, all such numbers must be divisible by 10.

10 is the sum of the numbers in the first Pythagorean triple: {3, 4, 5}.

10−2 is the smallest noncototient, 10+2 is the smallest nontotient, the average of them is 10.

10 is the average of the smallest nontotient (12) and the smallest noncototient (X).

10 is the first sublime number, a number that has a perfect number (6) of divisors, and the sum of its divisors is also a perfect number (24), there are only two known such numbers (including 10 itself).

10 is the product of the first three factorials (not including 0! = 1).

10 is the smallest base b such that the number of trailing zeros in n! (the factorial of n) does not depend on one of the primes or prime powers dividing b.

The square of any prime >3 is congruent to 1 mod 10 (in fact, congruent to 1 mod 20).

The sum of any pair of twin primes (other than 3 and 5) is divisible by 10. (i.e. end with 0)

The product of any pair of twin primes (other than 3 and 5) is congruent to E mod 10. (i.e. end with E)

10 is the largest number which is a divisor of the sum of all pairs of twin primes other than (3, 5).

10 is the largest number which is a divisor of the sum of all but finitely many pairs of twin primes (assuming that there are infinitely many pairs of twin primes).

10 is the largest number which is a divisor of xy for all Pythagorean triples {x, y, z}.

10 is the largest base such that both "all squares end with square digits" and "all primes not dividing the base end with either prime digits not dividing the base or 1" are true.

10 is the largest base such that all these three properties are true:

Property of the numerical system Property of the base (b) Bases (Note: 1 cannot be the base of the system)
All squares end with square digits Numbers n such that all x^2 mod n are squares 1, 2, 3, 4, 5, 8, 10, 14
All primes not dividing the base end with either prime digits not dividing the base or 1 (unit) Numbers n such that reduced residue system of n consists of only primes and 1 1, 2, 3, 4, 6, 8, 10, 16, 20, 26
All squares of the primes not dividing the base end with 1 Numbers n with the property m^2 == 1 (mod n) for all integer m coprime to n 1, 2, 3, 4, 6, 8, 10, 20 (exactly the divisors of 20)

10 is the largest base such that both "all squares end with square digits" and "all primes end with prime digits or 1" are true.

10 is the largest number n with the property that every number m in the range n < m < 2n that is coprime to n is also prime.

10 is the largest number n with the property that every number m in the range 1 < m ≤ 2n that is coprime to n is also prime.

10 is the largest natural number n such that for all even numbers k dividing n, both k−1 and k+1 are noncomposite (including 0).

10 is the largest natural number n such that for all numbers k dividing n, both k−1 and k+1 are either prime or power of 2 (or 0).

10 is the largest n such that all even divisors >2 of n are average of a twin prime pair.

10 is the largest number k such that k+j is prime for every j, where 1 <= j < k and gcd(j,k) = 1.

10 is the only number n ≤ 200000 such that n±1, n/2±1 and n/3±1 are all primes. (the next such n is 220990, note that this number contains 3 distinct digits all twice)

10!+1 is divisible by 121 = 11^2, and 10 is the only known non-semiprime n such that n!+1 is divisible by (n+1)^2.

10 is the smallest integer b such that more than one prime factor p of b attains the maximum of (p-1)*v_p(b) where v_p(b) is the valuation of b at p. (the other such b≤100 are 39, 68, 76 and 100)

10 is the smallest base n for which there exists k≤n (in fact, k≤n2) such that k! (where ! is the factorial) is not harshad number base n (k=7).

The Euclid number n#+1 (where # is the primorial) is prime for all n≤10 (but not for n=11).

Regular 10-gon is the highest polygon in the tessellation.

The numbers n ≤ 10 with terminate reciprocal are {1, 2, 3, 4, 6, 8, 9, 10}, and the divisors of 10 are {1, 2, 3, 4, 6, 10}, the difference of these two sets are 8 and 9 (of course, the second set is a subset of the first set), (8 is because it has more prime factors 2 than 10, and 9 is because it has more prime factors 3 than 10) (thus, the numbers of digits of the reciprocal of all these n except 8 and 9 are all 1, while the numbers of digits of the reciprocal of 8 and 9 are 2), and the numbers 8 and 9 are in the Catalan's conjecture (i.e. 8 and 9 are the only case of two consecutive perfect powers), besides, the product of 8 and 9 is 60, which is the smallest Achilles number, besides, the concatenation of 8 and 9 is 89, which is the smallest Ziesel number and the smallest integer such that the factorization of

$ x^n-1 $ over Q includes coefficients other than

$ \pm 1 $ (i.e. the 89th cyclotomic polynomial,

$ \Phi_{89} $ , is the first with coefficients other than

$ \pm 1 $ ), besides, the squares of 8 and 9 are the only two 2-digit automorphic numbers, besides, 8 and 9 are the only two natural numbers n such that centered n-gonal numbers (the kth centered n-gonal number is n×Tk+1, where Tk is the kth triangular number) cannot be primes (8 is because all centered 8-gonal numbers are square numbers (4-gonal numbers), 9 is because all centered 9-gonal numbers are triangular numbers (3-gonal numbers) not equal to 3, but all square numbers and all triangular numbers not equal to 3 are not primes, in fact, all polygonal numbers with rank > 2 are not primes, i.e. all primes p cannot be a polygonal number (except the trivial case, i.e. each p is the second p-gonal number)), assuming the Bunyakovsky conjecture is true. (i.e. 8 and 9 are the only two natural number n such that

$ \frac{n}{2}x^2+\frac{n}{2}x+1 $ is not irreducible) (Note that for n = 10, the centered 10-gonal numbers are exactly the star numbers)

If k=10, then k * prod{ p | k } = 60, which is the smallest Achilles number.

In base 10 (dozenal, the numeral system of this wiki), by Midy's theorem, if 0<a<p and the period of the dozenal representation of a/p is 2n, so that

$ \frac{a}{p}=0.\overline{a_1a_2a_3\dots a_na_{n+1}\dots a_{2n}} $

then the digits in the second half of the repeating dozenal period are the Es complement of the corresponding digits in its first half (e.g. 1/5 = 0.24 97, 1/7 = 0.186 X35, and 24 + 97 = EE, 186 + X35 = EEE), and the Es complement of the digits are:

  • 0 & E (the zero digit & the largest digit)
  • 1 & X (the unit digit & the only digit whose reciprocal is neither terminating nor purely repeating)
  • 2/3 & 8/9 (the prime factors of 10 & the difference of "the numbers n ≤ 10 with terminate reciprocal" and "the divisors of 10" (also the only digits whose reciprocal has exactly 2 digits, also the numbers in the Catalan's conjecture, also the only two natural numbers n such that centered n-gonal numbers (the kth centered n-gonal number is n×Tk+1, where Tk is the kth triangular number) cannot be primes, see above))
  • 4/6 & 5/7 (the only composite proper factors of 10 & the only primes < 10−1 not dividing 10)

(note that E is the largest number which is not sum of two composites, and E is also the smallest number >3 which is not sum of two primes) (if you regard 1 as prime, then E is the smallest number >1 which is not sum of two primes)

(also, note that the only primes < 10−1 not dividing 10 are 5 and 7 (and the value of 5+7 is exactly 10, and (5,7) are twin primes, and all sums of twin primes ≥(5,7) are also divisible by 10, and 10 is the largest number with this property), and 1/5 = 0.24 97, 1/7 = 0.186 X35, all digits except 0 and E (the zero digit and the largest digit) appear in the periodic part of either 1/5 or 1/7 (but not both) and exactly once in this number (1/5 or 1/7))

Except 0 and 1, 10 is the only natural number whose square is a Fibonacci number. (i.e. 0, 1 and 100 are the only three square Fibonacci numbers)

If (and only if) there are no Wall-Sun-Sun primes, then 10 is the largest n such that the Pisano period of n2 is the same as the Pisano period of n (both of them are 20).

The Frampton-Kephart primes are the primes p such that p−1 (or

$ \phi(p) $ , where

$ \phi $ is Euler's totient function) divides 10, these primes are exactly "1 + (the even divisors of 10)" plus the only even prime (2), they (and the number 1) are also the k such that such that d(k) < 3 and d(k − 1) = Pi(k) (where d(k) the divisor count function and Pi(k) the prime counting function), they (and the number 1) are also the possible solutions of x to U(x) = ±1 for some Lucas sequence, they are also the prime factors of 63X00, which is the largest number k satisfying that “for any positive integers x,y coprime to k, x^x == y (mod k) iff y^y == x (mod k)”, they are also the prime factors of 95900, the number of vertices of the 20-dimensional Leech lattice. (Note that 95900 =

$ \prod_{p \text{ prime },\ p-1|10}(\text{ if } p|10 \text{ then } p^{6-p},\ \text{ otherwise } p) $ ), and the product of these primes is 16E6, which is the denominator of B10 (where Bn is the nth Bernoulli number), (note that B10 is also the first Bernoulli number with even index whose numerator has a prime factor pn+3 (p=497) (i.e. 10 is the smallest number n such that there exists prime p such that (p, n) is an irregular pair)) besides, the dozenal representation of 1/p (with p over these primes), is terminate when p divides 10 (i.e. p = 2 or 3), and for the other three values of p, they are 0.2497, 0.186X35, and 0.0E, the period of them uses all the digits 0, 1, 2, ..., E exactly once.

The numbers n have this property: “all primes not divide n are congruent to 1 or −1 mod n” are {1, 2, 3, 4, 6}, they are exactly the proper divisors of 10. (and the LCM of them is 10)

The set of the proper divisors of 10 is the complete set of k in N such that

$ 2\cos\frac{2\pi}{k} $ is in Z.

10 is the largest natural number n such that the two sets are completely the same: {a | 0<=a<n, either a=1 or a is prime not dividing n} and {a | 0<=a<n, n+a is prime}. (for n=10, the two sets are both {1, 5, 7, E}, both of the two sets are exactly the natural numbers <10 coprime to 10) (the natural numbers n such that the two sets are completely the same are 1, 2, 4, 6, and 10, all of them are divisors of 10)

A number k satisfying that “for any positive integers x,y coprime to k, x^x == y (mod k) iff y^y == x (mod k)” if and only if k is divisible by

$ \lambda(k) $ (where

$ \lambda(n) $ is the Carmichael lambda function and k is a divisor of 63X00, note that 63X00 is the largest k such that

$ \lambda(k)=20 $ , and for all these k’s,

$ \lambda(k) $ also divides 20, besides, the prime factors of these numbers are exactly the Frampton-Kephart primes, i.e. the primes p such that p−1 (or

$ \phi(p) $ , where

$ \phi $ is Euler's totient function) divides 10. (equivalently, p−1 divides 20, since there is no prime p such that p−1 divides 20 but p−1 does not divide 10 (both 9 and 21 are composite (9=32, 21=52, both of them are squares of primes)), in fact, the primes p such that p−1 divides 10 (also 20) are exactly "1 + (the even divisors of 10)" plus the only even prime (2).

14060 is the smallest number divisible by all natural numbers from 1 to 10, it is also the smallest number cannot be written in primorial base using only the dozenal digits (i.e. the digits 0, 1, 2, ..., E) (for factorial base, the smallest such number is 1145000000).

10 is the largest known even number expressible as the sum of two primes in only one way (5+7).

10 is the smallest (and the only known) n>2 such that there exist n consecutive integers with n divisors (at least start with 507711EE945E5199770213X692X2X3, there are 11 consecutive integers with 10 divisors). (The only other two possible such n are 20 and X0)

All numbers ≤10 are panconsummate numbers, but the next number (11) is not, besides, the square of 10 (100) is conjectured to be the largest panconsummate number divisible by either 4 or 6 (i.e. not only for divisible by 10).

10 is the smallest n>1 such that the n-th cyclotomic polynomial has no root mod p for all primes pn.

Consider the prime race mod q (where q≥2) between qn+1 and qn-1. 10 is conjectured to be the n such that the q where qn+1 first takes lead over qn-1 is largest. (such q is known for all n<=1000 except n=10 and n=20)

10 is the smallest even number n such that there exists a prime pn+3 (p = 497) such that p divides the numerator of Bn (where Bn is the nth Bernoulli number). (i.e. 10 is the smallest number n such that there exists prime p such that (p, n) is an irregular pair) (note that the 10th prime (31) is also the smallest irregular prime)

10^2−1 (=EE) is the only number which is product of twin primes but not brilliant number, it is also the smallest composite n coprime to 10 (i.e. not divisible by 2 or 3) such that Fibonacci(n-1) is congruent to (1 - Legendre(n,5))/2 modulo n.

10 is the smallest even number n such that “n^k−1 and n^k+1 are always both prime or both composite for every integer k≥1” is conjectured to be true. (for odd number n, n^k−1 and n^k+1 are both even, and hence both composite (if n^k>3))

The ratio of the first two weird numbers (598/5X = E.E39310...) is very close to 10.

The only two known base 10 Wieferich primes are 1685 and 5E685, both of them end with 685, and it is conjectured that all base 10 Wieferich primes end with 685.

The sum of this 4 fifth powers is 10X, the Xth power of 10 (i.e. 10,000,000,000): 235 + 705 + 925 + E15 = 10X, this is a counterexample of Euler's sum of powers conjecture, since 10X equals 1005, it is also a fifth power, but it only require 4 fifth powers (less than the conjectured 5) to add. (Note that 100 is the smallest number whose fifth power is the sum of 4 or less fifth powers)

10 is the smallest weight for which a cusp form exists. This cusp form is the discriminant Δ(q) whose Fourier coefficients are given by the Ramanujan τ-function and which is (up to a constant multiplier) the 20th power of the Dedekind eta function. This fact is related to a constellation of interesting appearances of the number 10 in mathematics ranging from the value of the Riemann zeta function at −1 i.e. ζ(−1) = −1/10, the fact that the abelianization of SL(2,Z) has 10 elements, and even the properties of lattice polygons.

10 is the smallest admirable number.

10 is a Pell number, a pronic number, and a pentagonal number (5-gonal number).

10 is the number of pentominoes (5-ominoes).

1/10 = 10% is the property such that a Rubik's Cube is solvable.

10 is the least common multiple of 3 and 4, the number of sides of the first two regular polygons (equilateral triangle (3-gon) and square (4-gon)), 10 is also the least common multiple of 4 and 6, the number of faces of the first two regular polyhedrons (tetrahedron (4-hedron) and cube (6-hedron)).

The centered 10-gonal numbers (centered dozagonal numbers) are exactly the star numbers. (thus, the star numbers are exactly the numbers obtained as the concatenation of a triangular number followed by a 1)

The known generalized Fermat primes (primes of the form b2n+1) with bases b≤10 are: (this number can be prime only if b is even (since if it is odd then b2n+1 is always even, and thus can’t be prime), thus we only consider even bases)

base (b) n
2 0, 1, 2, 3, 4
4 0, 1, 2, 3
6 0, 1, 2
8 (impossible since 8=23)
X 0, 1
10 0

Except the case b=8 (which has algebraic factors: 82n+1 = (22n+1) × (42n−22n+1)), the n for these b are exactly (the nonnegative integers ≤ m for m = 4 to 0) in order, besides, "if b2n+1 is composite, then b2n+1+1 is also composite" is conjectured to be true for all even b≤10 (but not for b = next even number (12)).

There is a known generalized Cullen prime for all bases b ≤ 10 (but not for b = 11). (no matter whether you require nb−1 or not)

There is a known generalized Woodall prime for all bases b ≤ 100 (but not for b = 101). (no matter whether you require nb−1 or not)

A 10-sided polygon is a dozagon. A 10-faced polyhedron is a dozahedron. Regular cubes (hexahedrons, 6-faced) and octahedrons (8-faced) both have 10 edges, while regular octadozahedrons (18-faced) have 10 vertices.

The evenish numbers (numbers coprime to 10 and 10's digit is even) are exactly the numbers n such that

$ \left(\frac{n}{20}\right)=1 $ , and the oddish numbers (numbers coprime to 10 and 10's digit is odd) are exactly the numbers n such that

$ \left(\frac{n}{20}\right)=-1 $ . (thus, there are no oddish numbers that are squares)

Two dozagons (10-gons) and one triangle can fill a plane vertex, all solutions using at least one dozagon are {3, 10, 10}, {4, 6, 10}, {3, 3, 4, 10}, and {3, 4, 3, 10}, but only the first two solutions can fill the plane.

(there are 19 solutions for filling a plane vertex, but only E of them can fill the plane, the solutions are:

{3, 7, 36}, {3, 8, 20}, {3, 9, 16}, {3, X, 13}, {3, 10, 10}, {4, 5, 18}, {4, 6, 10}, {4, 8, 8}, {5, 5, X}, {6, 6, 6}

{3, 3, 4, 10}, {3, 4, 3, 10}, {3, 3, 6, 6}, {3, 6, 3, 6}, {3, 4, 4, 6}, {3, 4, 6, 4}, {4, 4, 4, 4}

{3, 3, 3, 3, 6}, {3, 3, 3, 4, 4}, {3, 3, 4, 3, 4}

{3, 3, 3, 3, 3, 3}

bold for the solutions that can fill the plane, note that dozagon is the highest regular polygon in convex uniform tiling, i.e. 10 is the largest bold number in the list)

10 and 18,E27,099,E93,490,727,709,9E9,1X4,841,59E,264,E0X,X13,59X,268,567,863,258,910,E74,98X,682,454 (=(2X6)(251 − 1)(227 − 1)(217 − 1)(27 − 1)(25 − 1)(23 − 1)) are the only two known sublime numbers. (A sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number, i.e. a positive integer n is sublime number if and only if both

$ \sigma_0(n) $ and

$ \sigma_1(n) $ are perfect numbers (where

$ \sigma_k(n) $ is the sum of the kth powers of the divisors of n, i.e.

$ \sigma_k(n)=\sum_{d\mid n} d^k $ ), e.g. for n=10 we have

$ \sigma_0(10)=1^0+2^0+3^0+4^0+6^0+10^0=6 $ and

$ \sigma_1(10)=1^1+2^1+3^1+4^1+6^1+10^1=24 $ , and both 6 and 24 are perfect numbers, thus 10 is a sublime number)

Although 6 is a divisor of 10, there exists a group of order 10 (A4) without a subgroup with order 6, it is the smallest such example (i.e. 10 is the smallest number n such that there exists k dividing n and a group of order n such that this group has no subgroup with order k)

All orders of non-solvable groups (thus all orders of non-cyclic simple groups) are divisible by either 10 or 18, and all orders of non-solvable groups ≤ 14000 are divisible by 10, (the smallest order of non-solvable groups not divisible by 10 is 14X28) the first two orders of non-cyclic simple groups are 50 and 120, and the greatest common divisor of them is indeed 10.

All odd perfect numbers (if exist) end with 1, 09, 39, 69, or 99, and if an odd perfect number ends with 1 (i.e. = 1 mod 10), then it has at least 10 distinct prime factors.

10 is the kissing number in three dimensions (for a long time, people did not know that whether the answer is 11, this was the subject of a famous disagreement between mathematicians Isaac Newton and David Gregory). (In two demensions, it is 6 = 10÷2, and in four dimensions, it is 20 = 10×2. The values of the kissing numbers are only known in 1, 2, 3, 4, 8 and 20 dimensions (note that all such numbers of dimensions are divisors of 20), and the kissing numbers in these numbers of dimensions are in order 2, 6, 10, 20, 180 and 95900, and the 8 dimension case and the 20 dimension case are in order the E8 lattice and the Leech lattice, however, there are also the upper bounds and the lower bounds for all other number of dimensions ≤ 20, see the list below, the dimensions in which the kissing number is known are listed in boldface)

Dimension Lower
bound
Upper
bound
1 2
2 6
3 10
4 20
5 34 38
6 60 66
7 X6 E2
8 180
9 216 264
X 358 3X2
E 406 606
10 5X0 951
11 802 1,245
12 E1X 1,X13
13 1,598 2,996
14 2,600 4,30E
15 3,116 6,4X8
16 4,346 9,710
17 6,210 12,438
18 X,0X0 19,338
19 14,060 27,708
1X 24,X60 3E,798
1E 45,XX6 60,000
20 95,900

(note that k(1)=2 divides k(2)=3, k(2)=3 divides k(3)=10, k(3)=10 divides k(4)=20, and the three quotients are all primes, however, k(4)=20 does not divide k(5), since 34≤k(5)≤38)

10 is the difference of "the largest number n such that x2 + x + n is prime for all 0≤x≤n−2" (35) and "the largest number n such that 2x2 + n is prime for all 0≤x≤n−1" (25).

10 is the difference of "the smallest number n such that (define an: a0 = 1, for k > 0, ak = (1+a02+a12+...+ak−12)/k) an is not integer" (37) and "the smallest number n such that ʃ0∞(cos(x)cos(x/2)cos(x/3)...cos(x/n)) ≠ π/2" (27).

10 is the difference of the first two numbers not of the form pm×qn with p, q primes, m≥0, n≥0 (36−26).

10 is the number of Latin squares of order 3.

10 is the smallest number such that it is unknown whether there are n−1 mutually orthogonal Latin squares of order n. (Note that there cannot be n mutually orthogonal Latin squares of order n, if n > 1)

10 is the smallest possible order of magic square with 1 and consecutive primes starting with 3. (the smallest possible order of magic square with consecutive primes starting with 3 is 2E)

10 appears in this form of these three almost integers related to the three largest Heegner numbers (117, 57 and 37):

  • $ e^{\pi\sqrt{37}}=10^3(9^2-1)^3+520-4.74596469\mathcal{E}07\ldots\times 10^{-4} $
  • $ e^{\pi\sqrt{57}}=10^3(19^2-1)^3+520-3.\mathcal{E}\mathcal{E}15385\mathcal{X}402\ldots\times 10^{-6} $
  • $ e^{\pi\sqrt{117}}=10^3(173^2-1)^3+520-6.82\mathcal{X}19322127\ldots\times 10^{-10} $

(the numbers with scientific notation are rounded to 10 significant figures)

All of them contains 103, besides, the three negative exponents of 10 of them are −10, −6 and −4 (the negative exponent of 10 of the largest number (eπ√117) is −10, etc.), the absolute values of them are exactly 10/n for n = 1, 2 and 3, besides, the numbers of the consecutive E around the dozenal point are also 10, 6 and 4 (exactly 10/n for n = 1, 2 and 3), since all of the integer part of these three numbers end with 51E (the reason is all of them are a multiple of 103 (=1000) plus the number 520 minus a number <1, see the formulas above), which already has one E before the dozenal point.

In music, 10 is the number of pitch classes in an octave (an octave has 10 semitones), not counting the duplicated (octave) pitch. Also, the total number of major keys, (not counting enharmonic equivalents) and the total number of minor keys (also not counting equivalents). This applies only to 10 tone equal temperament (10-TET), the most common tuning used today in western influenced music.

10 is used for timekeeping, e.g. one year has 10 months, one day has 20 hours, one hour has 50 minutes, and one minute has 50 seconds (all of these numbers are multiples of 10). Also, there are 10 signs of the zodiac (astrological signs), and the Chinese use a 10-year cycle for time-reckoning called Earthly Branches (Chinese zodiac).

10 is the number of numbers on a clock.

10 is the number of yellow stars on the Flag of Europe.

10 is the largest possible points for rolling two dices.

10 appears in the natural logarithm of Levy’s constant:

$ \frac{\pi^2}{10\ln 2} $ .

1/10 (the reciprocal of 10) also appears in the limit of Glaisher-Kinkelin constant:

$ A=\lim_{n\rightarrow\infty} \frac{K(n+1)}{n^{n^2/2+n/2+1/10} e^{-n^2/4}} $ .

The series 1 + 2 + 3 + 4 + ... is divergent, however, we can use the Riemann zeta function to define its sum: −1/10 (= −0.1), since zeta(−1) = −0.1 (thus, 10 + 20 + 30 + 40 + ... (the sum of all positive multiples of 10) = −1).

0 = Aries, 1 = Taurus, 2 = Gemini, 3 = Cancer, 4 = Leo, 5 = Virgo, 6 = Libra, 7 = Scorpio, 8 = Sagittarius, 9 = Capricorn, X = Aquarius, E = Pisces. (4k = fire, 4k+1 = earth, 4k+2 = air, 4k+3 = water; 3k = cardinal, 3k+1 = fixed, 3k+2 = mutable; 2k = yang, 2k+1 = yin)

0 = C, 1 = C#/Db, 2 = D, 3 = D#/Eb, 4 = E, 5 = F, 6 = F#/Gb, 7 = G, 8 = G#/Ab, 9 = A, X = A#/Bb, E = B. ({k, k+4, k+7} = k major chord, {k, k+3, k+7} = k minor chord, {k, k+4, k+8} = k augmented triad, {k, k+3, k+6} = k diminished triad, {k, k+4, k+7, k+E} = k major seventh chord, {k, k+3, k+7, k+X} = k minor seventh chord, {k, k+3, k+7, k+E} = k minor major seventh chord, {k, k+4, k+7, k+X} = k dominant seventh chord, {k, k+3, k+6, k+9} = k diminished seventh chord, {k, k+3, k+6, k+X} = k half-diminished seventh chord)

The digits in 1/5 = {2,4,7,9}, plus the digit 0 (the smallest digit) is the musical notes in the C-major pentatonic scale, plus the digit E (the largest digit) is the musical notes in the G-major pentatonic scale.

The digits in 1/7 = {1,3,5,6,8,X}, plus the digit 0 (the smallest digit) is the musical notes in the C#/Db-major diatonic scale, plus the digit E (the largest digit) is the musical notes in the F#/Gb-major diatonic scale.

(note that (C,F#/Gb) and (G,C#/Db) are tritones, and tritone is equal to 6 semitones, and 6 is equal to half of 10)

0 = unison, 1 = minor second, 2 = major second, 3 = minor third, 4 = major third, 5 = perfect fourth, 6 = tritone, 7 = perfect fifth, 8 = minor sixth, 9 = major sixth, X = minor seventh, E = major seventh, 10 = octave.

(they are the elements in the cyclic group Z10)

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