Problems & Puzzles: Puzzles

 Puzzle 958. P^3 has K tri-prime-partitions. Carlos Rivera asks this: Find the cube of a prime P with a large quantity of distinct tri-prime-partitions. This is my best solution: For P=1041701, P^3 = 1130392437132795101 has K=12 distinct tri-prime-partitions: 11 30392437 132795101 11 303924371 32795101 11 303924371327 95101 113 0392437 132795101 113 03924371327 95101 113039 2437 132795101 113039 24371 32795101 113039 24371327 95101 113039 243713279 5101 113039243 7 132795101 113039243 71 32795101 113039243 71327 95101 Q. Send your larger/best solution.

Contributions came from Pierandrea Formusa and Paul Cleary

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Pierandrea wrote on June 26, 2019:

My best solution is P=144013 , so that P^3 = 2986792777010197
has K=11 distinct tri-prime-partitions:
2 98679277 7010197
29 8679277 7010197
29867 9277 7010197
29867 927770101 97
298679 277 7010197
298679 27770101 97
29867927 7 7010197
29867927 770101 97
29867927 7701019 7
2986792777 0101 97
2986792777 01019 7

***

Paul wrote on June 28, 2019

I couldn't find a prime that gave more than 12 solutions, Here are the primes and the corresponding set of tri-prime partitions that I found.

47553353

{107,533413,902943370317977}

{107,5334139029433,70317977}

{107,5334139029433703,17977}

{10753,3413,902943370317977}

{10753,3413902943370317,977}

{1075334139029,433,70317977}

{1075334139029,433703,17977}

{1075334139029,43370317,977}

{10753341390294337,03,17977}

{10753341390294337,0317,977}

{10753341390294337031,7,977}

{10753341390294337031,797,7}

129237637

{2,158574419780719,322233853}

{2,15857441978071932223,3853}

{215857,4419780719,322233853}

{215857441,97,80719322233853}

{215857441,9780719322233,853}

{2158574419,7,80719322233853}

{2158574419,780719,322233853}

{2158574419,7807193,22233853}

{2158574419,78071932223,3853}

{2158574419,780719322233,853}

{21585744197,807193,22233853}

{21585744197,8071932223,3853}

132644027

{23,337,97095695163708447683}

{23,33797095695163,708447683}

{233,37,97095695163708447683}

{233,37970956951637,08447683}

{233,3797095695163708447,683}

{2333,7,97095695163708447683}

{2333,7970956951,63708447683}

{2333,797095695163,708447683}

{2333797,0956951,63708447683}

{2333797,095695163,708447683}

{233379709,56951,63708447683}

{233379709,5695163,708447683}

240031153

{13,829383937,197008759618577}

{13829383,937,197008759618577}

{13829383,9371,97008759618577}

{13829383,93719,7008759618577}

{138293839,37,197008759618577}

{138293839,3719,7008759618577}

{138293839,37197008759,618577}

{138293839,3719700875961857,7}

{1382938393,7,197008759618577}

{1382938393,71,97008759618577}

{1382938393,719,7008759618577}

{1382938393,719700875961857,7}

344986459

{41,05879003719,8751975100579}

{410587,9003719,8751975100579}

{410587,900371987,51975100579}

{410587,90037198751,975100579}

{4105879,003719,8751975100579}

{4105879003,71,98751975100579}

{4105879003,719,8751975100579}

{4105879003,71987,51975100579}

{4105879003,71987519,75100579}

{41058790037,19,8751975100579}

{41058790037,1987,51975100579}

{41058790037,1987519,75100579}

352964891

{43,973853,613181475155199971}

{439,738536131814751551,99971}

{439,73853613181475155199,971}

{4397,3853,613181475155199971}

{4397,38536131814751551,99971}

{43973,853,613181475155199971}

{43973,8536131814751,55199971}

{43973,8536131814751551999,71}

{43973853613181,47,5155199971}

{43973853613181,4751,55199971}

{43973853613181,475155199,971}

{43973853613181,4751551999,71}

651881831

{2,77017,132968034867732579191}

{2,7701713,2968034867732579191}

{2,770171329,68034867732579191}

{277,017,132968034867732579191}

{277,0171329,68034867732579191}

{277,017132968034867,732579191}

{27701,7,132968034867732579191}

{27701,71,32968034867732579191}

{27701,71329,68034867732579191}

{2770171,3,2968034867732579191}

{2770171329680348677,3,2579191}

{2770171329680348677,32579,191}

745562777

{41,4431398956138693035191,433}

{41443,1398956138693,035191433}

{414431,3989,56138693035191433}

{414431,39895613,8693035191433}

{414431,3989561386930351,91433}

{41443139,89,56138693035191433}

{41443139,895613,8693035191433}

{41443139,895613869303519,1433}

{4144313989561,3,8693035191433}

{4144313989561,38693,035191433}

{41443139895613,8693,035191433}

{41443139895613,86930351,91433}

1113550027

{13,8079497,6503359237833869683}

{13,8079497650335923783,3869683}

{13807,9497,6503359237833869683}

{13807,9497650335923,7833869683}

{138079,4976503359237833,869683}

{13807949,7,6503359237833869683}

{1380794976503,35923,7833869683}

{1380794976503,359237833,869683}

{13807949765033,5923,7833869683}

{13807949765033,5923783,3869683}

{13807949765033,59237833,869683}

{1380794976503359,23,7833869683}

Here are also the smallest primes when cubed that will partition in to 2 to 12 partitions.

59

{2,5,379}

{2,53,79}

137

{2,5,71353}

{2,571,353}

{257,13,53}

2633

{182537,7,137}

{182537,701,37}

{182537,7013,7}

{182537701,3,7}

6247

{2,43789,231223}

{2,43789231,223}

{2437,89,231223}

{2437,8923,1223}

{2437,89231,223}

50909

{13,19421,93239429}

{131,9421,93239429}

{131,94219,3239429}

{1319,421,93239429}

{1319,4219,3239429}

{1319,42193,239429}

287783

{2,3833,916130677687}

{2,383391613067,7687}

{2383,3,916130677687}

{2383,39161,30677687}

{2383,391613,677687}

{23833,9161,30677687}

{238339,1613,677687}

25763

{17,997,31869947}

{17,9973,1869947}

{17,99731869,947}

{1709,97,31869947}

{1709,97318699,47}

{17099,7,31869947}

{17099,73,1869947}

{17099,731869,947}

32999

{359,337,33098999}

{359,3373,3098999}

{3593,37,33098999}

{3593,373,3098999}

{3593,3733,98999}

{35933,7,33098999}

{35933,73,3098999}

{35933,733,98999}

{35933,73309,8999}

5876839

{2,2969779190701,977719}

{2029,6977,9190701977719}

{2029,69779190701977,719}

{2029697,7,9190701977719}

{20296977919,7,1977719}

{20296977919,701,977719}

{20296977919,7019,77719}

{20296977919,7019777,19}

{2029697791907,19,77719}

{2029697791907,19777,19}

144013

{2,98679277,7010197}

{29,8679277,7010197}

{29867,9277,7010197}

{29867,927770101,97}

{298679,277,7010197}

{298679,27770101,97}

{29867927,7,7010197}

{29867927,770101,97}

{29867927,7701019,7}

{2986792777,101,97}

{2986792777,1019,7}

1041701

{11,30392437,132795101}

{11,303924371,32795101}

{11,303924371327,95101}

{113,392437,132795101}

{113,3924371327,95101}

{113039,2437,132795101}

{113039,24371,32795101}

{113039,24371327,95101}

{113039,243713279,5101}

{113039243,7,132795101}

{113039243,71,32795101}

{113039243,71327,95101}

And as a matter of interest here is the prime that when cubed and partitioned in to 5 partitions gave the most, that I could find in the allotted time.  It has 47 sets

5876839

{2,2,9697,7,9190701977719}

{2,2,969779190701977,7,19}

{2,29,69779,1907,1977719}

{2,29,69779190701977,7,19}

{2,2969,7,7,9190701977719}

{2,2969,7791907,19,77719}

{2,2969,7791907,19777,19}

{2,2969,779190701977,7,19}

{2,2969779,19,7,1977719}

{2,2969779,19,701,977719}

{2,2969779,19,7019,77719}

{2,2969779,19,7019777,19}

{2,2969779,1907,19,77719}

{2,2969779,1907,19777,19}

{2,2969779190701,97,7,719}

{2,2969779190701,977,7,19}

{2029,6977,919,7,1977719}

{2029,6977,919,701,977719}

{2029,6977,919,7019,77719}

{2029,6977,919,7019777,19}

{2029,6977,919070197,7,719}

{2029,69779,19,7,1977719}

{2029,69779,19,701,977719}

{2029,69779,19,7019,77719}

{2029,69779,19,7019777,19}

{2029,69779,1907,19,77719}

{2029,69779,1907,19777,19}

{2029697,7,919,7,1977719}

{2029697,7,919,701,977719}

{2029697,7,919,7019,77719}

{2029697,7,919,7019777,19}

{2029697,7,919070197,7,719}

{2029697,79,19,7,1977719}

{2029697,79,19,701,977719}

{2029697,79,19,7019,77719}

{2029697,79,19,7019777,19}

{2029697,79,1907,19,77719}

{2029697,79,1907,19777,19}

{2029697,7919,7,19,77719}

{2029697,7919,7,19777,19}

{2029697,7919,701,977,719}

{20296977919,7,197,7,719}

{20296977919,701,97,7,719}

{20296977919,701,977,7,19}

{20296977919,7019,7,7,719}

{2029697791907,19,7,7,719}

{2029697791907,197,7,7,19}

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