Small integers and some of their properties

The specialest numbers

'Numbers to know', and their properties and where they show up

the special est numbers, the special est shapes

todo: multiplication tables https://en.m.wikipedia.org/wiki/Algebraic_number#Examples https://en.m.wikipedia.org/wiki/Golden_ratio

focus on non negative integers but also a few small positive reals. no negatives, no complex

small algebraic numbers? and all roots for coefficients 1-5 (1-6?); too many; maybe only up to degree 2 (quadratic), mb only monic

https://en.m.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals https://en.m.wikipedia.org/wiki/Quadratic_irrational

? https://en.m.wikipedia.org/wiki/Closed-form_expression#Closed-form_number

you can get all the prime factors of the cells in 3dim and 4dim regular polytopes with 2 3 5 -- 7 is not needed. all of 2 3 4 5 6 are factors of the numbers of faces (and hence edges, because there are dual regular polytopes) of various regular 3-polytopes (platonic solids) (and they are the only non-trivial factors of such; if you count trivial factors then you also get 8 12 20 (later: wait what? did i mean 'prime' instead of 'non-trivial'; but then what about 6? did i mean factors with no more than exponent 1 of any prime?); note that the only prime factor here with multiplicity is 2, which goes up to 2^3, reminiscent of the chernoff, also the other very divisible number sequences; however in 4d we have a 600-cell, which has two 5s, but nothing with 2 3).

note that every number below 12 can be reached by some sum of 0 or 1 of each of 1 2 3 5 (another special fact of 12, and another argument for base 12; note that 6 has this property with 1 2 3, and 4 with 1 2, and 2 with 1).

so eliminate 7 from the list of most special numbers. note as special property of 12. note that properties of 3d and 4d convex regular polytopes can be understood without 7, and that these dimensions are special for regular polytopes. note special property of 4,6,8,12,20 wrt regular 3-polytopes, and note d&d dice bc fair dice .

so most special numbers are 1-12, with 1-5 as more special than most (and lots of composite ish special things about 6 as well) (and 1-3 most interesting of these eg cubic, and of course 1-2 most interesting of those eg quadratic https://en.wikipedia.org/wiki/Quadratic and binomial https://en.wikipedia.org/wiki/Binomial_(polynomial)). but MB include 7 as 'first boring number'! also 7 is round of e^2 ( and note 3 is round of e and of pi, and 6 is round of 2pi). 16 (2^4, 4^2) and 20 (4*5 also platonic solid) and 24 (very divisible, also 24-cell, also factorial) and 25 (5^2) and 30 (very divisible (wait is it? it is primorial though); product of first 3 primes 2 3 5) and the primes in there, 13 17 19 what else? gets left out, or put in a section at the end, wherein we skip over boring numbers. note the other maxs (4 6 etc) in very divisible sequences. in misc section also include some more schn/ca numbers eg 360, and powers of primes (8 16 9 27 etc).

4 is the number of sides of a 3-simplex

other popular/fundamental Oeis sequences. also, cite the paper on the oeis frequency distribution to argue that squares and very divisible numbers are important.

note more primes than squares

prime factorization, very divisable number sequences, regular polytopes (most symmetric) (in general, properties of n-space). also lists of squares, primes, powers of small primes, rounded powers of e.

in every dimension, there is a hypercubical regular uniform space filling tiling, but in 2d and 4d, there are a few (2?) others too

another recitation of what is special about every integer through 12:

• 0: 0 element of ring. semantically: low, nothingness, out-of-band, semaphor, yang
• 1: first counting number. identity element of group. semantically: high, somethingness, primitive generator, whole, yin
• 2: first number of more than one object. semantically: number, duality/balance/symmetry
• 3: first prime other than 2. Sum of 1+2. The number of convex, regular polytopes in 5 and higher dimensions.
• 4: first composite. semantically: more involved symmetry. Last dimension with an irregular number of convex, regular polytopes. Number of faces and edges in a tetrahedron (3-simplex), a 3-D convex regular polytope (Platonic solid) (for each higher integer n, this integer is the number of vertices of the (n+1)-simplex, a higher-dimensional analog of a tetrahedron, so i won't list those).
• 5: first prime after a composite. first square pyramidal number after 1. the only prime that is both the sum and the difference of two primes [1]. Untouchable number; "If the Goldbach conjecture is true, then there are no other odd untouchable numbers" [2]. there are 5 platonic solids (convex regular polytopes in 3-D). 5-D is the first dimension in which there are exactly 3 convex regular polytopes, a pattern which will repeat for all higher dimensions. First Fibonacci number after a gap. Semantically: will. A_5 or A5, the group of even permutations on 5 elements, is the smallest non-solvable group, and the smallest non-cyclic simple group, and the smallest non-abelian simple group [3], and is the group of rotational/orientation-preserving symmetries of Dodecahedrons or Icosahedrons (their full symmetry group is A5 × Z2). S5 is the first "non-solvable" symmetric group. S5's non-solvability is related to "the non-existence of a general formula to solve quintic polynomials by radicals" [4].
• 6: first composite made up of two different primes. Primorial, Factorial (and so order of the symmetry group S3), and last number to be both primorial and factorial. first primorial after a gap. sum of 1+2+3. product of 1*2*3. product of 2*3. There are six convex regular polytopes in 4-D. The number of message bits in one block of ternary Golay code. Number of faces in a cube, a 3-D convex regular polytope (Platonic solid), and vertices in its dual, regular Octahedron. Number of edges in a tetrahedron, a 3-D convex regular polytope (Platonic solid). S6, the symmetric group on 6 elements, has an outer automorphism, unlike all other symmetric groups [5].
• 7: Various 'dischorant' properties; eg 7 is the only dimension, besides the familiar 3, in which a vector cross product can be defined. "the smallest number of sides of a regular polygon that cannot be constructed with straightedge and compass alone" [6].
• 8: first number (other than 0 and 1) of the from x^y where x != y; first number with three factors/first number of the form x*x*x (where x is other than 0 or 1); first number where 2 and 3 are mixed by exponentiation. Number of vertices in a cube, a 3-D convex regular polytope (Platonic solid), and faces in its dual, regular Octahedron. Number of cells in a 4-D hypercube, which is a 4-D convex regular polytope, and number of vertices in its dual, a 16-cell (or 4-orthoplex) (also a 4-D convex regular polytope). Fibonacci number.
• 9: first non-prime odd number. second number of the form x*x; there is a pleasing symmetry between the fact that this is the second number of that form, and the fact that the second prime, 3, is the value of x here; namely, on the one hand there is a bunch of 2s in these abstractions (x is repeated twice; this is the second number of this form; 3 is the second prime), but on the other hand the point is that there are no 2 factors in here, only 3s. Also, first non-prime odd number.
• 10: 'dischordant' in that the factors are 2 and 5 but unlike 2 and 3, these primes are not next to each other, either in terms of counting or in terms of the sequence of primes. Sum of 1+2+3+4. First even number after 2 that does not correspond to the number of faces in a Platonic solid. Number of faces and edges in a 4-simplex, a 4-D convex regular polyhedron. Number of vertices in a 5-orthoplex, a 5-D convex regular polyhedron (and number of 4-faces in its dual, a 5-D hypercube).
• 11: first prime after an odd-gap (through 7, all of the odd numbers were primes)
• 12: first number after a gap in the Chernoff sequence (superprimorials; cumulative product of primorials). first number with three prime factors that are not all the same. superior highly composite (SHCN) and colossally abundant; first of either of those sequences with a composite divisor. First of either of those sequences with a repeated factor. first superfactorial after a gap (12 = 1!*2!*3!). The smallest weight for which a cusp form exists. out of the even numbers: first highly composite number after a gap; first number that corresponds to a platonic solid after a gap; first practical number after a gap. Second composite SHCN or colossally abundant number; second SHCN or colossally abundant number after a gap; also 2 times (the first SHCN and colossally abundant number which is either composite or after a gap). The number of message bits in one block of binary Golay code. Also, i argue that 12 is a better radix than 6 because 6 has factors 2,3 whereas 12 has 2,2,3 (in other words, to address 6 exactly locations you need a bit and a trit, whereas for 12 you need two bits and a trit); but it's useful to divide by 2 much more often than it is to divide by 3, and having those 2 2s lets you divide by 4 too which is also useful, so it makes sense to have more factors of 2 than factors of 3 (more bits than trits). The number of faces in a regular Dodecahedron and the number of vertices in its dual, a regular Icosahedron, both of which are 3-D convex regular polytopes (Platonic solids). Number of edges in a cube, a 3-D convex regular polytope, and its dual, a regular Octahedron. Order of the rotational (orientation-preserving) symmetry group of a regular tetrahedron, a 3-D convex regular polytopes (Platonic solid). Number of vertices in a 6-orthoplex, a 6-D convex regular polyhedron (and number of 5-faces in its dual, a 6-D hypercube).

note: you can see here that the 'claim to fame' of any number after 6 is less solid; 8,9,10,12 all rely on more complex properties that only seem special because we already think that 2, 3, 4, compositeness (which we might identify with 4), and having lots of divisors (which we might identify with 6) are special. 7 and 11 are identified with absences and gaps and dischord (although so is 5). One might argue that 4, 5 and 6 share the same problem, and so the only truly interesting numbers are 0,1,2,3. Or, one might argue that 4 gets in because it is the first number whose specialness relies on the specialness of numbers below it (the first composite), and then 5 gets in as the first 'boring' number. Or one might argue that we should at least go up to 7 because it is the last odd number such that all odds leq it are prime, and because 6 is pretty interesting (the first SHCN and colossally abundant (CAN) after a gap), even if 5 is a little boring; or up until 9 because it demonstrates the first odd number that isn't prime. Personally, i think 0-3,0-4,0-5,0-6,0-7,0-12 have the best arguments. I lean towards 0-12 because i think a good radix should have lots of divisors, so 2, 6 or 12 are the obvious choices; 2 is less than 3, an obviously interesting number, so that's out; 6 is less than 10 so, relative to base 10, base 6 may be seen as 'inefficient' and unlikely to be accepted as a radix in our society. One can also make the argument that, even if only 0 thru 6 are truly interesting, it's a useful exercise to reflect on 7-12 because they are like the 'second coming' of 1 thru 6 (7 is like 1: very odd, centered, reaching; 8 is like 2: very even, filled with 2s; 9 is like 3, the 'second coming' of various numbers based on 2s, also it is filled with 3s; 10 is like 4, notably only for being the first to be made up of others in various ways (note that i think the real second coming of 4 is 16, however; also, 10 has some things in common with 6, as it is 1+2+3+4, so 10 is actually kind of an oddball); 11 is like 5, prime/very odd/reaching and the first such after some kind of gap; 12 is like 6, having many factors (SHCN and colossally abundant), and fittingly, is also = 6*2. By reflecting on 0 thru 12, you get some practice seeing how some of the fundamental themes expressed in 0 thru 7 play out, and you see some phenomena which might serve as counterexamples to hypotheses you might have held after only seeing 0 thru 7 (for example, you see that not all odd numbers are prime). So, as a plan of study, i like 0 thru 12.

Of course it would be nice to go up to 13 (the first prime whose odd-gap between it and the preceeding prime is smaller than the previous gap, also a standard torus can be sliced into 13 pieces with just 3 plane cuts, also a Fibonacci number), 20 (the number of faces in a regular Icosahedron which is the 3-D convex regular polytope with the most faces, and the number of vertices in its dual, a regular Dodecahedron which is a 3-D convex regular polytope, and the number of faces in a 5-simplex, a 5-D convex regular polytope), 24 (a factorial (and so order of the symmetry group S4 (which is also the symmetry group of a regular Tetrahedron, a a 3-D convex regular polytope (Platonic solid))), a multiple of 12, the first number with 4 prime factors that are not all the same), 30 (primorial (the first number with three distinct prime factors), the number of edges in a regular Dodecahedron or its dual, Icosahedron, which are 3-D convex regular polytopes, the number of faces in a 4-D hypercube, which is a 4-D convex regular polytope, and number of edges in its dual, a 16-cell (4-orthoplex); number of cells and vertices in a 24-cell, which is a 4-D convex regular polytope), or 60 (a multiple of 12 which is also a good radix (SHCN, CAN), and is the order of the smallest non-solvable group (A_5) which is also the smallest non-cyclic simple group [7] and which is the group of rotational/orientation-preserving symmetries of Dodecahedrons or Icosahedrons (their full symmetry group, however, is A5 × Z2, of order 120), and number of edges in a 6-orthoplex), or 71, which is the largest prime factor found in the order of any of the sporadic simple groups (sporadic simple groups are 26 or 27 (depending on how you classify things) exceptional groups found in the classification of finite simple groups [8]). But that's way too many to get through. Given the short duration of a human life, learning a little bit about each of the first 12 positive integers is probably all the time that can be spared, except for people who want to specialize in this sort of thing.

(I stopped at 12 instead of 13 because i think 12 would be a nice radix, and also because in some ways primes are the LEAST interesting number, having as they do the simplest possible prime factorization, so the increased prime gap between 7 and 11 followed by the smaller gap between 11 and 13 might be looked at as something interesting happening at 9, followed by ordinary boring stuff at 13, rather than as something interesting happening at 13; but in any case already told you what i consider to be the most interesting things about 13 in this section, so in a way i included 13 after all).

Some properties that we'll cover

Factors

primes

very divisible numbers: SHCN, CA first 15 same todo chart, todo enter these into entries chart below, also intersection of HCN and SA

factorials: 2 6 24 120

squares from 2 to 12: 4 9 16 25 36 49 64 81 100 121 144

cubes from 2 to 7: 8 27 64 125 216 343

n^n from 2 to 4: 4 9 256

primorials from 2 to 210: 2 6 30 210

todo more squares than primes: 1/n vs 1/log(n) [9]