Problems & Puzzles: Puzzles

Puzzle 595. p*q -> p^2 + p*q + q^2 JM Bergot sent the following nice puzzle: From a semiprime p*q get p^2 + p*q + q^2... Let's tart with the semiprime 10=2*5 2*5--->2^2 + 10 + 5^2=39=3*13 3*13--->3^2 + 39 + 13^2=217=7*31 7*31--->49+217+961=1227=3*409 3*409--->9+1227+167,281=168,517=43*3919 43*3919--->43^2 + 168517 + 3919^2=15,528,927=3*5,176,309 ...   Q1. How far goes the semiprime 10? Q2. Can you get a larger chain?

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Contributions came from Jim Howell, Torbjörn Alm, Antoine Verroken & Hakan Summakoğlu

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All of them discovered that starting with the semiprime 10 you can go one more step:

3*5176309--->3^2 + 15528927 + 5176309^2=26794190392417=7*3827741484631

7*3827741484631--->7^2 + 26794190392417 + 3827741484631^2=14651604873191926199598627 = 3 * 67 * 211 * 345467092810637009257 (Not a semiprime).

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Antoine wrote:

Q2:  no better result for p up to 1300 and q < p.

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Torbjorn wrote:

I also submit 2 solutions where the 6th iterations ends:

a) 173*229
b) 2*32251

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