Little math problem(s) May 2021

[Valentin Albillo]

.
Hi, Pier:

I already posted that problem 15 years ago in this message:

Short & Sweet Math Challenge: Cooking Conjectures

My own solution was

1 DESTROY ALL @ OPTION BASE 0 @ K=150 @ DIM P(K)
2 FOR I=0 TO K @ P(I)=I*I*I*I*I @ NEXT I @ R=.2 @ L=2^R @ M=3^R @ N=4^R
3 FOR E=K TO 1 STEP -1 @ T=P(E) @ DISP E
4 FOR D=E-1 TO E/N STEP -1 @ F=T-P(D) @ U=F^R
5 FOR C=INT(U) TO U/M STEP -1 @ G=F-P(C) @ V=G^R @ FOR B=INT(V) TO V/L STEP -1
6 IF NOT FP((G-P(B))^R) THEN A=(G-P(B))^R @ DISP A;B;C;D;E @ END
7 NEXT B @ NEXT C @ NEXT D @ NEXT E

upon running, it eventually outputs:

>RUN

27 84 110 133 144

which is the sought-for counterexample, 13 seconds on Emu71, 90 min on a physical HP-71B.

Regards.
V.