A355278 Lower left of the Cayley table for the primes when made into a group using the bijection (2, 3, 5, 7, ...) -> (0, +1, -1, +2, ...) into (Z, +); read by rows.

2, 3, 7, 5, 2, 11, 7, 13, 3, 19, 11, 5, 17, 2, 23, 13, 19, 7, 29, 3, 37, 17, 11, 23, 5, 31, 2, 41, 19, 29, 13, 37, 7, 43, 3, 53, 23, 17, 31, 11, 41, 5, 47, 2, 59, 29, 37, 19, 43, 13, 53, 7, 61, 3, 71, 31, 23, 41, 17, 47, 11, 59, 5, 67, 2, 73, 37, 43, 29

Offset: 1

Author

M. F. Hasler, Sep 08 2022

Keywords

• nonn
• tabl

Comments

The primes can be given the structure of a group via a bijection with the group (Z, +). The simplest such bijection is to assign the integers (0, 1, -1, 2, -2, ...) to the primes, in increasing order, i.e., the composition of the prime counting function A000720, decreased by 1, with the canonical enumeration A001057.

Since this is an abelian group, the table (an infinite square matrix) is symmetric, T(m,n) = T(n,m), and it is sufficient to list only the lower left part, m >= n >= 1.

Each row and each column (of the complete square matrix) is a permutation of the primes, as always for Cayley tables. The inverse of a prime p, for the group operation *, is found at the top of the column in which 2 appears in the row starting with p, cf. 3rd formula.

Other simple bijections from the primes into (Z, +) would be to associate the negative integers to the primes congruent to 1 (mod 4) or to those congruent to 1 (mod 3), and the nonnegative integers to the others, in increasing order.

Examples

			The correspondence of primes p with integers m is as follows:
   p |  2 |  3 |  5 |  7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | ...
  ---+----+----+----+----+----+----+----+----+----+----+----+----
   m |  0 |  1 | -1 |  2 | -2 |  3 | -3 |  4 | -4 |  5 | -5 | ...
The table of the abelian group (P, *) based on this correspondence with (Z, +) starts as follows:
  [ 2  3  5  7 11 13 17 ...]
  [ 3  7  2 13  5 19 11 ...]
  [ 5  2 11  3 17  7 23 ...]
  [ 7 13  3 19  2 29  5 ...]
  [11  5 17  2 23  3 31 ...]
  [13 19  7 29  3 37  2 ...]
  [17 11 23  5 31  2 41 ...]
  [...    ...    ...    ...]
This means that 3 * 3 = 7, 3 * 5 = 2, 3 * 7 = 13, etc.: for example, primes 3 and 7 are associated to integers 1 and 2, so 3 * 7 is the prime associated to the integer 1 + 2 = 3, which yields 13.
Since 2, associated to 0 in Z, is the neutral element in the group (P, *), the first row and column is identical to the list of the primes themselves. Therefore we do not need to show row and columns headings in addition to the first row & column of the body of the table.
Since the table is symmetric,  T(m,n) = T(n,m)  <=>  p * q = q * p, the sequence lists only the lower left part: 2; 3, 7; 5, 2, 11; 7, 13, 3, 19; ...
The list of inverses of the primes 2, 3, 5, 7, 11, ... with respect to this group operation is 2, 5, 3, 11, 7, 17, 13, ... = A000040(A065190(n)), i.e., after the initial 2, list the primes with the two members of each subsequent pair swapped: swap 3 and 5, 7 and 11, 13 and 17, etc.
		

Links

• Antonio (u/hilberts12th), An example of a binary operation * such that (Set of prime numbers, *) is a group?, in r/abstractalgebra on reddit.com, May 18 2022

Crossrefs

Cf. A000040, A000720, A001057.

Programs

PARI

A355278(m,n) = inv(f(prime(m))+f(prime(n)))
inv(x, p)=while(!mapisdefined(INV,x,&p), mapput(INV, f(p=prime(#INV+1)), p)); p
INV=Map(); f(p)=(p=primepi(p))\2*(-1)^(p%2)

Formula

T(m, n) = T(n, m), so only m >= n >= 1 is listed: row m has length m.

T(n, n) = A000040(A042948(n)) = A000040(A014601(n-1)+1)

T(m, 1) * T(k, 1) = 2 (neutral element of the group operation *) <=> T(k, 1) is the inverse of T(m, 1) <=> T(m, k) = 2.