Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1258 42312143

 From the always intgeresting Prime Curios! pages of my friend G. L. Honaker,Jr., I stole, once more, the following curio:

42312143 is the largest prime that can be split into two 4-digit primes, (4231, 2143), each consisting of the first four positive digits. [Loungrides]

Q1. Find ALL the primes of ten digits composed of two times the digits 1, 3, 5, 7, 9 such that it can be split into two 5-digits distinct primes, each consiting of the first five odd positive digits (1, 3, 5, 7 and 9). The order of the integers doesn't matter in any of the three primes.

Q2. Find the smallest and the largest prime of eleven decimal digits (0-9; not necessarily distinct) that can be split into two 5 & 6 digits primes such that the set of the digits of both of these two 5 & 6 digits primes is equal to the set of the digits of the 11 digits initial prime. The order of the integers doesn't matter in any of the three primes.  


From Feb 14-20, 2026 contributions came from Michael Branicky, Gennady Gusev, Jeff Heleen, Rahid Zaman, Emmanuel Vantieghem and Ankit Chakraborty.

***

Michael wrote:

For Q1, I found 164 such primes:
1359715937 13597 15937
1359753719 13597 53719
1359757193 13597 57193
1359775913 13597 75913
1375913597 13759 13597
1375915973 13759 15973
1375939157 13759 39157
1375953791 13759 53791
1573913597 15739 13597
1573937951 15739 37951
1573957139 15739 57139
1573957193 15739 57193
1593737591 15937 37591
1593753197 15937 53197
1593753719 15937 53719
1593779153 15937 79153
1593779531 15937 79531
1597317539 15973 17539
1597357193 15973 57193
1597371593 15973 71593
1597373951 15973 73951
1597375193 15973 75193
1597375931 15973 75931
1597395731 15973 95731
1735919753 17359 19753
1735953917 17359 53917
1735975193 17359 75193
1735991573 17359 91573
1735995731 17359 95731
1753915937 17539 15937
1753953791 17539 53791
1753957193 17539 57193
1753979153 17539 79153
1753979531 17539 79531
1753991573 17539 91573
1975315973 19753 15973
1975331957 19753 31957
1975337951 19753 37951
1975353197 19753 53197
1975375931 19753 75931
1975391573 19753 91573
1975395731 19753 95731
3195715973 31957 15973
3195737591 31957 37591
3195751973 31957 51973
3195753719 31957 53719
3195775931 31957 75931
3195795731 31957 95731
3715953719 37159 53719
3715957139 37159 57139
3715975391 37159 75391
3759117359 37591 17359
3759151973 37591 51973
3759157139 37591 57139
3759171593 37591 71593
3759175193 37591 75193
3759175391 37591 75391
3759179531 37591 79531
3759191753 37591 91753
3759195317 37591 95317
3795171593 37951 71593
3795179153 37951 79153
3795195317 37951 95317
3915713597 39157 13597
3915731957 39157 31957
3915775931 39157 75931
3915791573 39157 91573
5197317539 51973 17539
5197337591 51973 37591
5197339157 51973 39157
5197375193 51973 75193
5197375913 51973 75913
5197379153 51973 79153
5197395317 51973 95317
5319713759 53197 13759
5319739157 53197 39157
5319751973 53197 51973
5319771593 53197 71593
5319773951 53197 73951
5319775931 53197 75931
5319791573 53197 91573
5319795317 53197 95317
5371915937 53719 15937
5371957193 53719 57193
5371975913 53719 75913
5379113759 53791 13759
5379115739 53791 15739
5379117539 53791 17539
5379137591 53791 37591
5379153917 53791 53917
5379175391 53791 75391
5379195713 53791 95713
5391713759 53917 13759
5391731957 53917 31957
5391753791 53917 53791
5713915973 57139 15973
5713937591 57139 37591
5713937951 57139 37951
5713951973 57139 51973
5713953197 57139 53197
5713975913 57139 75913
5713975931 57139 75931
5719313759 57193 13759
5719315739 57193 15739
5719331957 57193 31957
5719375391 57193 75391
5719375913 57193 75913
5719379153 57193 79153
7135915739 71359 15739
7135995731 71359 95731
7159357193 71593 57193
7159373951 71593 73951
7159375193 71593 75193
7159379153 71593 79153
7159391753 71593 91753
7395117539 73951 17539
7395153917 73951 53917
7395179153 73951 79153
7519313597 75193 13597
7519317359 75193 17359
7519317539 75193 17539
7519353791 75193 53791
7519371359 75193 71359
7519371593 75193 71593
7519391573 75193 91573
7519391753 75193 91753
7519395713 75193 95713
7519395731 75193 95731
7539113597 75391 13597
7539117539 75391 17539
7539171359 75391 71359
7591339157 75913 39157
7593153719 75931 53719
7593157139 75931 57139
7593195317 75931 95317
7915337159 79153 37159
7915351973 79153 51973
7915353197 79153 53197
7915395713 79153 95713
7953113759 79531 13759
7953117539 79531 17539
7953153917 79531 53917
7953157139 79531 57139
7953175931 79531 75931
9157331957 91573 31957
9157337951 91573 37951
9157357193 91573 57193
9157375391 91573 75391
9157379531 91573 79531
9175315739 91753 15739
9175337159 91753 37159
9175375931 91753 75931
9175391573 91753 91573
9531715937 95317 15937
9531719753 95317 19753
9531751973 95317 51973
9531757139 95317 57139
9531775193 95317 75193
9571315973 95713 15973
9571337159 95713 37159
9571357139 95713 57139
9571375391 95713 75391
9573157193 95731 57193
9573171593 95731 71593

For Q2, the smallest found was 1000991009 and the largest was 9997197919.
If the two parts are constrained to have the same multiset of digits, I found 1003910093 and again 9997197919.

For Q1, 28 of them are emirps:
1375939157 13759 39157 1375953791 13759 53791 1573913597 15739 13597 1593753197 15937 53197 1593753719 15937 53719 1735953917 17359 53917 1735975193 17359 75193 1753915937 17539 15937 1753953791 17539 53791 1753957193 17539 57193 1753991573 17539 91573 3715953719 37159 53719 3715957139 37159 57139 3759117359 37591 17359 3759179531 37591 79531 3759195317 37591 95317 3915791573 39157 91573 7135915739 71359 15739 7159391753 71593 91753 7519353791 75193 53791 7519395713 75193 95713 7519395731 75193 95731 7593195317 75931 95317 7953175931 79531 75931 9175315739 91753 15739 9531719753 95317 19753 9571315973 95713 15973 9571357139 95713 57139
Sure. indeed these two have all 3 5 digits primes emirps:
1375939157 13759 39157
7519395731 75193 95731
***

Gennady wrote:

Q1.
min=1359715937
max=9573171593
There are 164 numbers in total in applied file.
Q2.
min=10235914867 (102359 & 14867) 
max=98754165203 (987541 & 65203)
provided that all 10 digits are used.
Did I understand correctly that ALL 10 digits should be used? (Yes!, CR)

***

Jeff wrote:

Hola Carlos,
Q1. There seem to be 164 solutions:
1359715937, 1359753719, 1359757193, 1359775913, 1375913597, 
1375915973, 1375939157, 1375953791, 1573913597, 1573937951, 
1573957139, 1573957193, 1593737591, 1593753197, 1593753719, 
1593779153, 1593779531, 1597317539, 1597357193, 1597371593, 
1597373951, 1597375193, 1597375931, 1597395731, 1735919753, 
1735953917, 1735975193, 1735991573, 1735995731, 1753915937, 
1753953791, 1753957193, 1753979153, 1753979531, 1753991573, 
1975315973, 1975331957, 1975337951, 1975353197, 1975375931, 
1975391573, 1975395731, 3195715973, 3195737591, 3195751973, 
3195753719, 3195775931, 3195795731, 3715953719, 3715957139, 
3715975391, 3759117359, 3759151973, 3759157139, 3759171593, 
3759175193, 3759175391, 3759179531, 3759191753, 3759195317, 
3795171593, 3795179153, 3795195317, 3915713597, 3915731957, 
3915775931, 3915791573, 5197317539, 5197337591, 5197339157, 
5197375193, 5197375913, 5197379153, 5197395317, 5319713759, 
5319739157, 5319751973, 5319771593, 5319773951, 5319775931, 
5319791573, 5319795317, 5371915937, 5371957193, 5371975913, 
5379113759, 5379115739, 5379117539, 5379137591, 5379153917, 
5379175391, 5379195713, 5391713759, 5391731957, 5391753791, 
5713915973, 5713937591, 5713937951, 5713951973, 5713953197, 
5713975913, 5713975931, 5719313759, 5719315739, 5719331957, 
5719375391, 5719375913, 5719379153, 7135915739, 7135995731, 
7159357193, 7159373951, 7159375193, 7159379153, 7159391753, 
7395117539, 7395153917, 7395179153, 7519313597, 7519317359, 
7519317539, 7519353791, 7519371359, 7519371593, 7519391573, 
7519391753, 7519395713, 7519395731, 7539113597, 7539117539, 
7539171359, 7591339157, 7593153719, 7593157139, 7593195317, 
7915337159, 7915351973, 7915353197, 7915395713, 7953113759, 
7953117539, 7953153917, 7953157139, 7953175931, 9157331957, 
9157337951, 9157357193, 9157375391, 9157379531, 9175315739, 
9175337159, 9175375931, 9175391573, 9531715937, 9531719753, 
9531751973, 9531757139, 9531775193, 9571315973, 9571337159, 
9571357139, 9571375391, 9573157193, 9573171593
 

 

***

Rahid wrote:

For Q1, there's 164 solutions, listed below.
For Q2, I wasn't sure whether it had to be strictly (5 digits)|(6 digits) or if it could also be (6 digits)|(5 digits), but

min: 10007|107717
max: 99991|119191

if we also allow (6)|(5), then
min: 100003|30133
max: 999983|98993

[solved using Python script]

Thank you,
Rahid Zaman.

1359715937 = 13597|15937
1359753719 = 13597|53719
1359757193 = 13597|57193
1359775913 = 13597|75913
1375913597 = 13759|13597
1375915973 = 13759|15973
1375939157 = 13759|39157
1375953791 = 13759|53791
1573913597 = 15739|13597
1573937951 = 15739|37951
1573957139 = 15739|57139
1573957193 = 15739|57193
1593737591 = 15937|37591
1593753197 = 15937|53197
1593753719 = 15937|53719
1593779153 = 15937|79153
1593779531 = 15937|79531
1597317539 = 15973|17539
1597357193 = 15973|57193
1597371593 = 15973|71593
1597373951 = 15973|73951
1597375193 = 15973|75193
1597375931 = 15973|75931
1597395731 = 15973|95731
1735919753 = 17359|19753
1735953917 = 17359|53917
1735975193 = 17359|75193
1735991573 = 17359|91573
1735995731 = 17359|95731
1753915937 = 17539|15937
1753953791 = 17539|53791
1753957193 = 17539|57193
1753979153 = 17539|79153
1753979531 = 17539|79531
1753991573 = 17539|91573
1975315973 = 19753|15973
1975331957 = 19753|31957
1975337951 = 19753|37951
1975353197 = 19753|53197
1975375931 = 19753|75931
1975391573 = 19753|91573
1975395731 = 19753|95731
3195715973 = 31957|15973
3195737591 = 31957|37591
3195751973 = 31957|51973
3195753719 = 31957|53719
3195775931 = 31957|75931
3195795731 = 31957|95731
3715953719 = 37159|53719
3715957139 = 37159|57139
3715975391 = 37159|75391
3759117359 = 37591|17359
3759151973 = 37591|51973
3759157139 = 37591|57139
3759171593 = 37591|71593
3759175193 = 37591|75193
3759175391 = 37591|75391
3759179531 = 37591|79531
3759191753 = 37591|91753
3759195317 = 37591|95317
3795171593 = 37951|71593
3795179153 = 37951|79153
3795195317 = 37951|95317
3915713597 = 39157|13597
3915731957 = 39157|31957
3915775931 = 39157|75931
3915791573 = 39157|91573
5197317539 = 51973|17539
5197337591 = 51973|37591
5197339157 = 51973|39157
5197375193 = 51973|75193
5197375913 = 51973|75913
5197379153 = 51973|79153
5197395317 = 51973|95317
5319713759 = 53197|13759
5319739157 = 53197|39157
5319751973 = 53197|51973
5319771593 = 53197|71593
5319773951 = 53197|73951
5319775931 = 53197|75931
5319791573 = 53197|91573
5319795317 = 53197|95317
5371915937 = 53719|15937
5371957193 = 53719|57193
5371975913 = 53719|75913
5379113759 = 53791|13759
5379115739 = 53791|15739
5379117539 = 53791|17539
5379137591 = 53791|37591
5379153917 = 53791|53917
5379175391 = 53791|75391
5379195713 = 53791|95713
5391713759 = 53917|13759
5391731957 = 53917|31957
5391753791 = 53917|53791
5713915973 = 57139|15973
5713937591 = 57139|37591
5713937951 = 57139|37951
5713951973 = 57139|51973
5713953197 = 57139|53197
5713975913 = 57139|75913
5713975931 = 57139|75931
5719313759 = 57193|13759
5719315739 = 57193|15739
5719331957 = 57193|31957
5719375391 = 57193|75391
5719375913 = 57193|75913
5719379153 = 57193|79153
7135915739 = 71359|15739
7135995731 = 71359|95731
7159357193 = 71593|57193
7159373951 = 71593|73951
7159375193 = 71593|75193
7159379153 = 71593|79153
7159391753 = 71593|91753
7395117539 = 73951|17539
7395153917 = 73951|53917
7395179153 = 73951|79153
7519313597 = 75193|13597
7519317359 = 75193|17359
7519317539 = 75193|17539
7519353791 = 75193|53791
7519371359 = 75193|71359
7519371593 = 75193|71593
7519391573 = 75193|91573
7519391753 = 75193|91753
7519395713 = 75193|95713
7519395731 = 75193|95731
7539113597 = 75391|13597
7539117539 = 75391|17539
7539171359 = 75391|71359
7591339157 = 75913|39157
7593153719 = 75931|53719
7593157139 = 75931|57139
7593195317 = 75931|95317
7915337159 = 79153|37159
7915351973 = 79153|51973
7915353197 = 79153|53197
7915395713 = 79153|95713
7953113759 = 79531|13759
7953117539 = 79531|17539
7953153917 = 79531|53917
7953157139 = 79531|57139
7953175931 = 79531|75931
9157331957 = 91573|31957
9157337951 = 91573|37951
9157357193 = 91573|57193
9157375391 = 91573|75391
9157379531 = 91573|79531
9175315739 = 91753|15739
9175337159 = 91753|37159
9175375931 = 91753|75931
9175391573 = 91753|91573
9531715937 = 95317|15937
9531719753 = 95317|19753
9531751973 = 95317|51973
9531757139 = 95317|57139
9531775193 = 95317|75193
9571315973 = 95713|15973
9571337159 = 95713|37159
9571357139 = 95713|57139
9571375391 = 95713|75391
9573157193 = 95731|57193
9573171593 = 95731|71593

 

***

Emmanuel wrote:

Q1.
These are the desired primes : {1359715937,1359753719,1359757193,1359775913,1375913597,1375915973,1375939157,1375953791,1573913597,1573937951,1573957139,
1573957193,1593737591,1593753197,1593753719,1593779153,1593779531,1597317539,1597357193,1597371593,1597373951,1597375193,1597375931,1597395731,
1735919753,1735953917,1735975193,1735991573,1735995731,1753915937,1753953791,1753957193,1753979153,1753979531,1753991573,1975315973,1975331957,
1975337951,1975353197,1975375931,1975391573,1975395731,3195715973,3195737591,3195751973,3195753719,3195775931,3195795731,3715953719,3715957139,
3715975391,3759117359,3759151973,3759157139,3759171593,3759175193,3759175391,3759179531,3759191753,3759195317,3795171593,3795179153,3795195317,
3915713597,3915731957,3915775931,3915791573,5197317539,5197337591,5197339157,5197375193,5197375913,5197379153,5197395317,5319713759,5319739157,
5319751973,5319771593,5319773951,5319775931,5319791573,5319795317,5371915937,5371957193,5371975913,5379113759,5379115739,5379117539,5379137591.
,5379153917,5379175391,5379195713,5391713759,5391731957,5391753791,5713915973,5713937591,5713937951,5713951973,5713953197,5713975913,5713975931,
5719313759,5719315739,5719331957,5719375391,5719375913,5719379153,7135915739,7135995731,7159357193,7159373951,7159375193,7159379153,7159391753,
7395117539,7395153917,7395179153,7519313597,7519317359,7519317539,7519353791,7519371359,7519371593,7519391573,7519391753,7519395713,7519395731,
7539113597,7539117539,7539171359,7591339157,7593153719,7593157139,7593195317,7915337159,7915351973,7915353197,7915395713,7953113759,7953117539,
7953153917,7953157139,7953175931,9157331957,9157337951,9157357193,9157375391,9157379531,9175315739,9175337159,9175375931,9175391573,9531715937,
9531719753,9531751973,9531757139,9531775193,9571315973,9571337159,9571357139,9571375391,9573157193,9573171593}

Q2
Here, it was not entirely clear to me whether all of the ten digits should be used or not.
So, I solved both questions.
When pandigitallity is required :
Smallest : 10139248657
Biggest : 98717562403
In no pandigitallity is required :
Smallest : 10007100043
Biggest : 99991999169

***

On Feb. 20, 2026, Ankit Chakraborty, senet the following results, but Gmail sent his mail to Spam Box, so I couldn´t add them on time. Sorry...

Answer to Q1 : 

1359715937
1359753719
1359757193
1359775913
1375913597
1375915973
1375939157
1375953791
1573913597
1573937951
1573957139
1573957193
1593737591
1593753197
1593753719
1593779153
1593779531
1597317539
1597357193
1597371593
1597373951
1597375193
1597375931
1597395731
1735919753
1735953917
1735975193
1735991573
1735995731
1753915937
1753953791
1753957193
1753979153
1753979531
1753991573
1975315973
1975331957
1975337951
1975353197
1975375931
1975391573
1975395731
3195715973
3195737591
3195751973
3195753719
3195775931
3195795731
3715953719
3715957139
3715975391
3759117359
3759151973
3759157139
3759171593
3759175193
3759175391
3759179531
3759191753
3759195317
3795171593
3795179153
3795195317
3915713597
3915731957
3915775931
3915791573
5197317539
5197337591
5197339157
5197375193
5197375913
5197379153
5197395317
5319713759
5319739157
5319751973
5319771593
5319773951
5319775931
5319791573
5319795317
5371915937
5371957193
5371975913
5379113759
5379115739
5379117539
5379137591
5379153917
5379175391
5379195713
5391713759
5391731957
5391753791
5713915973
5713937591
5713937951
5713951973
5713953197
5713975913
5713975931
5719313759
5719315739
5719331957
5719375391
5719375913
5719379153
7135915739
7135995731
7159357193
7159373951
7159375193
7159379153
7159391753
7395117539
7395153917
7395179153
7519313597
7519317359
7519317539
7519353791
7519371359
7519371593
7519391573
7519391753
7519395713
7519395731
7539113597
7539117539
7539171359
7591339157
7593153719
7593157139
7593195317
7915337159
7915351973
7915353197
7915395713
7953113759
7953117539
7953153917
7953157139
7953175931
9157331957
9157337951
9157357193
9157375391
9157379531
9175315739
9175337159
9175375931
9175391573
9531715937
9531719753
9531751973
9531757139
9531775193
9571315973
9571337159
9571357139
9571375391
9573157193
9573171593

 

***

 

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