Problems & Puzzles: Puzzles

 Puzzle 1112 Concatenation of prime expansion in different bases Giorgos Kalogeropoulos sent the following nice puzzle:   We start with a prime and expand it in different bases. Then we concatenate those numbers. We are searching for the least prime such that the concatenated numbers are primes for all bases 2,3,4,5...   151 = 10010111 base 2 = 2^7+2^4+2^2+2^1+2^0 = 128+16+4+2+1 concatenate-> 12816421 is prime (note that if the coefficient of a power is zero, we skip this power)      151 = 12121 base 3 = 3^4+2*3^3+3^2+2*3^1+3^0 = 81+54+9+6+1 -> 8154961 is prime   151 = 2113 base 4 = 2*4^3+4^2+4^1+3*4^0 = 128+16+4+3 -> 1281643  not prime So, 151 is the least prime such that the concatenated numbers are primes in bases 2 and 3. 7=111base 2=2^2+2^1+2^0=4+2+1->421 is prime 7= 21base 3=2*3^1+3^0=6+1->61 is prime 7=13base 4=4^1+3*4^0=4+3->43 is prime 7=12 base 5=5^1+2*5^0=5+2->52 not prime   So, 7 is the least prime such that the concatenated numbers are primes in bases 2,3 and 4.   Q. Can you find the least primes that work for all bases 2,3,4,5,6,7... ?

During the week 19-25 November 2022, contributions came from Michael Branicky, Jean-Marc Rebert, Gennady Gusev, Emmanuel Vantieghem

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Michael wrote:

648133 = 10011110001111000101 base 2 = 2^19+2^16+2^15+2^14+2^13+2^9+2^8+2^7+2^6+2^2+2^0 = 524288+65536+32768+16384+8192+512+256+128+64+4+1 -> 52428865536327681638481925122561286441 is prime 648133 = 1012221001221 base 3 = 3^12+3^10+2*3^9+2*3^8+2*3^7+3^6+3^3+2*3^2+2*3^1+3^0 = 531441+59049+39366+13122+4374+729+27+18+6+1 -> 5314415904939366131224374729271861 is prime 648133 = 2132033011 base 4 = 2*4^9+4^8+3*4^7+2*4^6+3*4^4+3*4^3+4^1+4^0 = 524288+65536+49152+8192+768+192+4+1 -> 5242886553649152819276819241 is prime 648133 = 131220013 base 5 = 5^8+3*5^7+5^6+2*5^5+2*5^4+5^1+3*5^0 = 390625+234375+15625+6250+1250+5+3 -> 390625234375156256250125053 is prime 648133 = 21520341 base 6 = 2*6^7+6^6+5*6^5+2*6^4+3*6^2+4*6^1+6^0 = 559872+46656+38880+2592+108+24+1 -> 55987246656388802592108241 is not prime.
So, 648133 is the least prime such that the concatenated numbers are primes in bases 2, 3, 4 and 5.

5280878431 = 100111010110000111100111101011111 base 2 = 2^32+2^29+2^28+2^27+2^25+2^23+2^22+2^17+2^16+2^15+2^14+2^11+2^10+2^9+2^8+2^6+2^4+2^3+2^2+2^1+2^0 = 4294967296+536870912+268435456+134217728+33554432+8388608+4194304+131072+65536+32768+16384+2048+1024+512+256+64+16+8+4+2+1 -> 429496729653687091226843545613421772833554432838860841943041310726553632768163842048102451225664168421 is prime 5280878431 = 111122000220102100001 base 3 = 3^20+3^19+3^18+3^17+2*3^16+2*3^15+2*3^11+2*3^10+3^8+2*3^6+3^5+3^0 = 3486784401+1162261467+387420489+129140163+86093442+28697814+354294+118098+6561+1458+243+1 -> 348678440111622614673874204891291401638609344228697814354294118098656114582431 is prime 5280878431 = 10322300330331133 base 4 = 4^16+3*4^14+2*4^13+2*4^12+3*4^11+3*4^8+3*4^7+3*4^5+3*4^4+4^3+4^2+3*4^1+3*4^0 = 4294967296+805306368+134217728+33554432+12582912+196608+49152+3072+768+64+16+12+3 -> 429496729680530636813421772833554432125829121966084915230727686416123 is prime 5280878431 = 41303401102211 base 5 = 4*5^13+5^12+3*5^11+3*5^9+4*5^8+5^6+5^5+2*5^3+2*5^2+5^1+5^0 = 4882812500+244140625+146484375+5859375+1562500+15625+3125+250+50+5+1 -> 4882812500244140625146484375585937515625001562531252505051 is prime 5280878431 = 2232003315131 base 6 = 2*6^12+2*6^11+3*6^10+2*6^9+3*6^6+3*6^5+6^4+5*6^3+6^2+3*6^1+6^0 = 4353564672+725594112+181398528+20155392+139968+23328+1296+1080+36+18+1 -> 435356467272559411218139852820155392139968233281296108036181 is prime 5280878431 = 244602504013 base 7 = 2*7^11+4*7^10+4*7^9+6*7^8+2*7^6+5*7^5+4*7^3+7^1+3*7^0 = 3954653486+1129900996+161414428+34588806+235298+84035+1372+7+3 -> 395465348611299009961614144283458880623529884035137273 is not prime. So, 5280878431 is the least prime such that the concatenated numbers are primes in bases 2, 3, 4, 5, and 6

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JM Rebert wrote:

2 5 -> 1 primes [41]
4 7 -> 3 primes [421, 61, 43]
5 648133 -> 4 primes [52428865536327681638481925122561286441, 5314415904939366131224374729271861, 5242886553649152819276819241, 390625234375156256250125053]
6 5280878431 -> 5 primes [429496729653687091226843545613421772833554432838860841943041310726553632768163842048102451225664168421, 348678440111622614673874204891291401638609344228697814354294118098656114582431, 429496729680530636813421772833554432125829121966084915230727686416123, 4882812500244140625146484375585937515625001562531252505051, 435356467272559411218139852820155392139968233281296108036181]

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The least solution for base 2, 3, 4, 5 is 648133. No found primes for base 2 - 6 upto 1.3*10^9.

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Emmanuel wrote:

648133  is the least prime such that the concatenated numbers are primes in bases 2, 3, 4 and 5
5280878431  is the least prime such that the concatenated numbers are primes in bases 2, 3, 4, 5 and 6.
The least prime such that the concatenated numbers are primes in bases 2, 3, 4, 5, 6 and 7  must be > 14*10^10.

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