All numbers ≤4 divide 10 (the base of the numeral system of this wiki), but 5 does not. (note that 10 is the smallest such number)

All numbers ≤4 satisfy the property that general polynomial equation of degree n has a solution in radicals, but 5 does not. (all numbers ≥5 do not)

All numbers ≤4 satisfy the property that the Fermat number 2^(2^n)+1 is prime, but 5 does not. (all numbers from 5 to 28 do not)

All numbers ≤4 satisfy the property that the double Mersenne number 2^(2^(nth prime)−1)−1 is prime, but 5 does not. (all numbers from 5 to 15 do not)

All numbers ≤4 satisfy the property that the completed graph K_n is planar graph, but 5 does not. (all numbers ≥5 do not)

All numbers ≤4 satisfy the property that 1/n has terminated dozenal, but 5 does not. (only the 3-smooth numbers do)

All numbers ≤4 satisfy the property that the alternating group A_n is a solvable group, but 5 does not. (all numbers ≥5 do not)

All numbers ≤4 satisfy the property that the Lucas number L(2^n) is prime, but 5 does not. (all numbers from 5 to 20 do not)

10 (the base of the numeral system of this wiki) is the least common multiple of the numbers ≤4. (note that this is true only for “the set of the numbers ≤4”, this is true neither for “the set of the numbers ≤3” nor for “the set of the numbers ≤5”)

For the number 4! = 20:

If and only if n is a divisor of 20, then m² = 1 mod n for every integer m coprime to n.

If and only if n is a divisor of 20, then the Dirichlet characters mod n are all real.

If and only if n is a divisor of 20, then n is divisible by all numbers less than or equal to the square root of n.

If and only if n is a divisor of 20, then k−1 is prime for all divisors k>2 of n.

If and only if n+1 is a divisor of 20, then is squarefree for all 0 ≤ k ≤ n, i.e. all numbers in the nth row of the Pascal's triangle are squarefree (the topmost row (i.e. the row which contains only one 1) of the Pascal's triangle is the 0th row, not the 1st row). (Note that all such n are primes or 1 or 0, and 20 is the largest number m such that if n+1 is a divisor of m, then n is prime or 1 or 0, besides, if and only if m is a divisor of 20, then m satisfies this condition)

If we only have the numbers 1 to 20 (including 1 and 20), then only the primes dividing 10 (i.e. primes ≤3) can be squared, since 5^2 = 21 > 20, and for the numbers such that the reciprocal of n terminates, they can only have at most 2 digits (which is the case of 8, 9, 14, 16 and 20), since the numbers with terminate reciprocal with >2 digits, they must be divisible by either 2^5 = 28 or 3^3 = 23, but both are >20. (these (prime power) numbers are >20: (prime > 3)^(>1), (odd prime)^(>2), 2^(>4))

20 is the largest integer that is divisible by all natural numbers no larger than its square root.

The exponents on the right hand side of are exactly the numbers n such that 20n+1 is square. (note that 10 is one of such numbers)

20 is the Euler characteristic of a K3 surface.

20 is the order of the cyclic group equal to the stable 3-stem in homotopy groups of spheres: Template:Pi_n+3(Sⁿ) = Z/20Z for all n ≥ 5.

20 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 10, 20 — are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/20Z)^× = {±1, ±5, ±7, ±E} is isomorphic to the additive group (Z/2Z)³. This fact plays a role in monstrous moonshine.

The kissing number is only known in dimensions dividing 20 (1, 2, 3, 4, 6, 8, 10, 20) except 6 and 10

The densest packing is only known in dimensions dividing 20 (1, 2, 3, 4, 6, 8, 10, 20) but not the composite divisors of 10 (i.e. 4, 6, 10)

If xy≤20, then at least one of x and y is a divisor of 10 (this is not true for xy=21, 5×5=21, but 5 is not a divisor of 10).

googol (mod n) = googolplex (mod n) for all 1 ≤ n ≤ 20 (but not for n = 21).

20-cell is the only one of the six convex regular 4-polytopes which is not the four-dimensional analogue of one of the five regular Platonic solids (i.e. 20-cell is the only 4-dimension regular polytope which does not have a regular analogue in 3 dimensions).

For all numbers n ≤ 100 (but not for n = 101, and not for n = smallest prime > 100 (i.e. 105)), there is k ≤ 6 such that nk−1 or nk+1 (or both) is prime. (note that for n = 101, k = 7, 8 and 9 also not satisfy this condition, the smallest k satisfying this condition for n = 101 is X)

20 is the GCD of all Fermat-Wilson quotients. (thus, Fermat-Wilson quotients are never primes)

20 is the kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 20 such spheres form the cells of a 20-cell), for 3-dimensional space, the kissing number is 10, the centers of 10 such spheres form the faces of a 10-hedron.

20-cell is the only convex regular 4-polytope which does not have a regular analogue in 3 dimensions. (but with a regular analogue in 2 dimensions: regular hexagon, also, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dozahedron)

The exponents on the right hand side of are exactly the numbers n such that kn+1 is square for k=20.

20 is the smallest number n such that the graph {2≤x≤n, 2≤y≤n, x divides y} is not planar graph.

20! is very close to Avogadro constant, which is defined by 1/10 (10%) of the mass of one carbon-10 atom.

20 is the number of hours in a day, 20 is also the number of solar terms in a year.

20-karat gold is pure, 16-karat gold is 16 parts gold, 6 parts another metal (forming an alloy with 90% gold), 12-karat gold is 12 parts gold, X parts another metal (forming an alloy with 70% gold), and so forth.