Problems & Puzzles: Puzzles

 

 

Problems & Puzzles: Puzzles

Puzzle 1261 1124638579

From the always interesting pages of my friend G. L. Honaker, Jr., we know that:

1124638579 is the smallest zeroless pandigital prime whose square (i.e., 1264811933375139241) is also zeroless pandigital. [Gupta]

  
Now I (CR) am going to make a little change. I will ask for zeroless pandigital emirps whose square of both primes are also zeroless pandigitals.

The reverse of the prime 1124638579 is 9758364211 which unfortunateli is not a prime; moreover the square of 9758364211 is 95225672074525652521 which is not zeroless (it has one zero) neither pandigital (it lacks the digits 3 & 8). So 1124638579  is not a good example of "zeroless pandigital emirp whose squares of both primes are zeroless pandigitals."

Q1. Send the first ten zeroless pandigital emirps whose squares of both primes are zeroless pandigitals?

Q2. Afters receiving from several puzzlers the answer to Q1 with such ease, I add the following question: send the largest emirp example you may compute with the same  characteristics: zeroless pandigital emirp whose squares of both primes are zeroless pandigitals


From 7 to 13, March, 2026, contributions came from Emmanuel Vantieghem, Jeff Heleen, Michael Branicky, J. M. Rebert, Paul Cleary, Ankit Chakraborty, Gennady Gusev

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Emmanuel wrote:

Q1.

The smallest solutions are :1134872569, 1154269873, 1163859427, 1194857263, 1247568793, 
1263879457, 1269587443, 1274356829, 1274518639, 1325459687.

Q2.
This is my champion : 959871962436773939999989999979956948637695639792841  (51  digits) whose squares is
921354184272223561667292837185397338432487716927327443535792864722196113849573366229336817869394851281 (102 digits)

RevPrime is: 148297936596736849659979999989999939377634269178959 whose square is
21992277998849782635529539341711595929335364419817378366789752678291711327647621425328463323968323681 (101 digits

🥈

 

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 Jeff wrote:

For Q1:

1134872569 1287935747868659761
9652784311 93176244954687744721
 
1154269873 1332338939715436129
3789624511 14361253934371989121
 
1163859427 1354568765816768329
7249583611 52556462532879799321
 
1194857263 1427683878943851169
3627584911 13159372286514877921
 
1247568793 1556427893267476849
3978657421 15829714873678371241
 
1263879457 1597391281826614849
7549783621 56999232723919871641
 
1269587443 1611852275423278249
3447859621 11887735966122263641
 
1274356829 1623985327618935241
9286534721 86239727124338547841
 
1274518639 1624397761158412321
9368154721 87762322876594587841
 
1325459687 1756843381862137969
7869545231 61929742142754843361
 
For Q2:
The largest I found is:
9898968415327 97989575687641537568516929
7235148698989 52347376696482219329622121

***

Michael wrote:
Q1:
The first 10 terms are:
1134872569, 1154269873, 1163859427, 1194857263, 1247568793, 1263879457, 1269587443, 1274356829, 1274518639, 1325459687

Q2:
Here is one with 27 digits:
p = 111111111786259439111111111, prime and zeroless pandigital (zp)
r = 111111111934952687111111111, reverse(p), prime and zp
p^2: 12345679162378641245948721566627682615399136987654321, zp
r^2: 12345679195421585468838399111595874582356192987654321, zp

Here is one with 50 digits:
p = 11111111155714589263147738583211111111155714589263, prime and zeroless pandigital (zp)
r = 36298541755111111111238583774136298541755111111111, prime and zp
p^2 = 123456791114645195491518774984887772753822221443145697792569559199131685749558688235695344794883169, zp
r^2 = 1317584133547544822646488711464144516295565659912739456767561814586482961192872829168225288987654321, zp

Here is a pair with 73 digits each:
p = 1134872569211799111111486523799235914111111486523792394147897654231111111, prime and zeroless pandigital (zp)
r = 1111111324567987414932973256841111114195329973256841111119971129652784311, prime and zp
p^2 = 1287935748349389763722877176813452818439427151478179439954552729471664142482916632367823638912729914513533362723
911444348444766313879533627654321, zp
r^2 = 1234568375583227473565636276852879587288496681662224849286646727158965133813534896968387836885632199446921983155192611
311384224994441594687744721, zp

🥇


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Rebert wrote:

Q1.
Emirps p with p <= reverse(p):
1134872569, 1154269873, 1163859427, 1194857263, 1247568793, 1263879457, 1269587443, 1274356829, 1274518639, 1325459687.
p = 1134872569 is an emirp, and p = 1134872569, p^2 = 1287935747868659761, Rp = 9652784311 and Rp^2 = 93176244954687744721 are zeroless pandigital numbers.
p = 1154269873 is an emirp, and p = 1154269873, p^2 = 1332338939715436129, Rp = 3789624511 and Rp^2 = 14361253934371989121 are zeroless pandigital numbers.
p = 1163859427 is an emirp, and p = 1163859427, p^2 = 1354568765816768329, Rp = 7249583611 and Rp^2 = 52556462532879799321 are zeroless pandigital numbers.
p = 1194857263 is an emirp, and p = 1194857263, p^2 = 1427683878943851169, Rp = 3627584911 and Rp^2 = 13159372286514877921 are zeroless pandigital numbers.
p = 1247568793 is an emirp, and p = 1247568793, p^2 = 1556427893267476849, Rp = 3978657421 and Rp^2 = 15829714873678371241 are zeroless pandigital numbers.
p = 1263879457 is an emirp, and p = 1263879457, p^2 = 1597391281826614849, Rp = 7549783621 and Rp^2 = 56999232723919871641 are zeroless pandigital numbers.
p = 1269587443 is an emirp, and p = 1269587443, p^2 = 1611852275423278249, Rp = 3447859621 and Rp^2 = 11887735966122263641 are zeroless pandigital numbers.
p = 1274356829 is an emirp, and p = 1274356829, p^2 = 1623985327618935241, Rp = 9286534721 and Rp^2 = 86239727124338547841 are zeroless pandigital numbers.
p = 1274518639 is an emirp, and p = 1274518639, p^2 = 1624397761158412321, Rp = 9368154721 and Rp^2 = 87762322876594587841 are zeroless pandigital numbers.
p = 1325459687 is an emirp, and p = 1325459687, p^2 = 1756843381862137969, Rp = 7869545231 and Rp^2 = 61929742142754843361 are zeroless pandigital numbers.

Q2. I found: 
p = 9898968415327 (13 digits) is an emirp, and p = 9898968415327, p^2 = 97989575687641537568516929, Rp = 7235148698989 and Rp^2 = 52347376696482219329622121 are zeroless pandigital numbers.

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Paul wrote:

For Q1.

prime reversePrime square reverseSquare
1134872569 9652784311 1287935747868659761 93176244954687744721
1154269873 3789624511 1332338939715436129 14361253934371989121
1163859427 7249583611 1354568765816768329 52556462532879799321
1194857263 3627584911 1427683878943851169 13159372286514877921
1247568793 3978657421 1556427893267476849 15829714873678371241
1263879457 7549783621 1597391281826614849 56999232723919871641
1269587443 3447859621 1611852275423278249 11887735966122263641
1274356829 9286534721 1623985327618935241 86239727124338547841
1274518639 9368154721 1624397761158412321 87762322876594587841
1325459687 7869545231 1756843381862137969 61929742142754843361

Q2. Largest I could find

length 34 digits
prime 1699798879999994759799987892668883
reversePrime 3888662987899979574999999788979961
squares
2889316232449236585427776887971714764668969673185194266776676467689
15121699833463196694526942644245549895979563968593194529456859561521

🥉

***

Ankit wrote:

Q1:

1. 1134872569  (rev: 9652784311)  p^2=1287935747868659761  q^2=93176244954687744721
 2. 1154269873  (rev: 3789624511)  p^2=1332338939715436129  q^2=14361253934371989121
 3. 1163859427  (rev: 7249583611)  p^2=1354568765816768329  q^2=52556462532879799321
 4. 1194857263  (rev: 3627584911)  p^2=1427683878943851169  q^2=13159372286514877921
 5. 1274518639  (rev: 9368154721)  p^2=1624397761158412321  q^2=87762322876594587841
 6. 1357428691  (rev: 1968247531)  p^2=1842612651149973481  q^2=3873998343287595961
 7. 1458126739  (rev: 9376218541)  p^2=2126133586986774121  q^2=87913474128592168681
 8. 1482653917  (rev: 7193562841)  p^2=2198262637595442889  q^2=51747346347415991281
 9. 1641935287  (rev: 7825391461)  p^2=2695951486695772369  q^2=61236751517891714521
10. 1715843629  (rev: 9263485171)  p^2=2944119359179889641  q^2=85812157513336899241

5/10

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Gennady wrote:

Q1.
Prime, emirp, prime^2, emirp^2
1274518639, 9368154721, 1624397761158412321, 87762322876594587841
1134872569, 9652784311, 1287935747868659761, 93176244954687744721
1134872569, 9652784311, 1287935747868659761, 93176244954687744721
1134928567, 7658294311, 1288062852192673489, 58649471753894964721
1134928567, 7658294311, 1288062852192673489, 58649471753894964721
1357428691, 1968247531, 1842612651149973481, 3873998343287595961
1458126739, 9376218541, 2126133586986774121, 87913474128592168681
1482653917, 7193562841, 2198262637595442889, 51747346347415991281
1486925371, 1735296841, 2210947058923487641, 3011255126384579281
1154269873, 3789624511, 1332338939715436129, 14361253934371989121
 
6/10
Q2.
11998423726589 (14 digits), 98562732489911, 143962171922773866225574921, 9714612235877757433716787921
 

 

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