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

[Valentin Albillo]

       
Hi, all:
        
Continuing with my original solutions, today it's time for:

Tier 4 - The Challenge:

Consider the n-point dataset  (1, 2), (2, 3), (3, 5), (4, 7), (5, 11), (6, 13), ..., (n, 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))

Write a program that takes no inputs but computes and outputs the limit of the sum of the coefficients  a₀,  a₁, ... , a_n-1  when n tends to infinity.

My original solution:

My original solution for the HP-71B is this 4-liner (168 bytes):

       1  DESTROY ALL @ OPTION BASE 0 @ REPEAT @ N=N+1 @ DIM C(N) @ T=S
       2  FOR I=1 TO N @ C(I)=FPRIM(C(I-1)+1) @ NEXT I @ S=0
       3  FOR I=1 TO N-1 @ FOR J=N TO I+1 STEP -1 @ C(J)=C(J)-C(J-1) @ NEXT J @ NEXT I
       4  FOR I=1 TO N @ S=S+C(I)/FACT(I-1) @ NEXT I @ UNTIL S=T @ DISP N;S

       >RUN
              20    3.40706916561     { it converged to the limit after fitting the first 20 primes: 2, 3, 5, ..., 71) }

Notes:

• Line 1  initializes and begins the loop to compute the sum of the first n coefficients
• Line 2  fills an array with the first n primes
• Line 3  computes the forward differences in-place (replacing the primes)
• Line 4  computes the sum of the coefficients (differences / factorials) and loops back until it agrees with the previous sum, then outputs it

That's all for Tier 4, thanks a lot to Albert Chan for his interest in this particular tier and congratulations for providing a correct solution and some explanation but please, Albert, next time *do* provide actual code for an HP calculator of your choice, so that people can try your solution for themselves.

In the next days I'll post my solutions for the remaining tiers.

V.