[VA] Short & Sweet Math Challenge #24: "2019 Spring Special 5-tier"

[Albert Chan]

(03-21-2019 02:08 AM)Valentin Albillo Wrote:  Tier 4:  Advanced
[MRM: HP-11C and up]

Consider the n-point dataset (x_i , y_i) where  x_i = 1, 2, 3, 4, 5, 6, ..., n  (the natural numbers) and  y_i = 2, 3, 5, 7, 11, 13, ..., p_n  (the prime numbers), and the (n-1)^st degree polinomial fit to this dataset of the form:

            P(x) = a₀ + a₁ (x-1) + a₂ (x-1) (x-2) + ... + a_n-1 (x-1) (x-2) (x-3) ... (x-(n-1))

Getting sum of above coefficients can be done by doing forward difference of the primes:

2 3 5 7 11 13 17 19 23 29 ... ; primes
1 2 2 4 2 4 2 4 6 ... ; Δ
1 0 2 -2 2 -2 2 2 ... ; Δ²
-1 2 -4 4 -4 4 0 ... ; Δ³
3 -6 8 -8 8 -4 ... ; Δ⁴
...

a_k = Δ^k(0) / k!

Σ(a_k, k = 0 to Inf) = 2/0! + 1/1! + 1/2! - 1/3! + 3/4! + ...

Sum converge very fast:
2.0
3.0
3.5
3.33333 333333
3.45833 333333
3.38333 333333
3.41527 777778
3.40476 190476
3.40761 408730
3.40696 097884
3.40708 691578
3.40706 684905 ; 6 digits accuracy with 12 primes
3.40706 938140
3.40706 915834
3.40706 916344
3.40706 916625
3.40706 916552
3.40706 916564
3.40706 916563 ; 12 digits accuracy with 19 primes