The recursive irrationality of π

Extract

A primitive-recursive sequence of rational numbers s_n is said to be primitive-recursively irrational, if there are primitive recursive functions n(k), i(p, q) > 0 and N(p, q) such that:

1. (k)(n ≥ n(k) → ∣s_n – s_n(k)∣ < 2^−k).

2. (p)(q)(q > 0 & n ≥ N(p, q) → ∣s_n ± p/q∣ > 1/i(p, q)).

The object of the present note is to establish the primitive-recursive irrationality of a sequence which converges to π. In a previous paper we proved the primitive-recursive irrationality of the exponential series Σxⁿ/n!, for all rational values of x, and showed that a primitive-(general-) recursively irrational sequence s_n is strongly primitive-(general-)recursive convergent in any scale, where a recursive sequence s_n is said to be strongly primitive-(general-)recursive convergent in the scale r (r ≥ 2), if there is a non-decreasing primitive-(general-) recursive function r(k) such that

,

where [x] is the greatest integer contained in x, i.e. [x] = i if i ≤ x < i + 1, [x] = —i if i ≤ —x < i+1, where i is a non-negative integer.

A rational recursive sequence s_n is said to be recursive convergent, if there is a recursive function n(k) such that

.

If a sequence s_n is strongly recursive convergent in a scale r, then it is recursive convergent and its limit is the recursive real number where, for any k ≥ 0,

.