Problems & Puzzles: Puzzles

 Puzzle 636. Primorial as product of palindromes Following a puzzle-idea from the entry 915 from the always interesting Claudio Meller's site, here we will ask for primorials that are at the same time product of two or more palindromes (two or more digits, please).As a matter of fact I found a primorial that it is a product of two palindromes, & perhaps the simplest/smallest example: 17# = 510510 = 595 x 858 Q. Can you find more examples (no matter if you need more than two palindromic factors)?

Contributions came from Giovanni Resta, Jan van Delden, Emmanuel Vantieghem & Hakan Summakoğlu

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Giovanni wrote:

In my report "7#" means "prime(7)#" or "17#"

After your solution for 7#, there is one more solution for 8# and no more solutions exist from 9# to 38#.

Then solutions come back from 39# and on.

Here are the found solutions until 64# "obtained in one or two hours".

Usually there are several ways to write each product. Here I list just one solution.

(BTW, Giovanni sent his approach in order to solve this puzzle, but I will publish it later, in order to do not spoil by now your own fun).

8# = 494*595*33
39# = 325494523*7973797*13031*1491855581941*15151*10001*8582858*73337*535*101*33733*989*111
40# = 798323897*325494523*7973797*62296097079069226*151*15151*1253256523521*565*72427*11
41# = 14141*798323897*325494523*7973797*13031*1491855581941*15151*10001*8582858*73337*989*555
42# = 181*14141*798323897*325494523*7973797*13031*1491855581941*15151*10001*8582858*73337*989
*55
5
43# = 191*181*14141*798323897*325494523*7973797*13031*1491855581941*15151*10001*8582858*73337
*
989*555
44# = 36863*181*14141*798323897*325494523*7973797*13031*1491855581941*15151*10001*8582858
*73337*
989*555
45# = 1174711*36863*31313*14141*6081806*7973797*13031*151*3987337893*10001*1681861*7207027
*
1131311*515*76067
46# = 33233*1174711*36863*31313*972131279*147222741*13031*151*15151*989303989*2871782*535*949
*
76067*30503*12121
47# = 45154*9130319*1174711*36863*31313*96199399169*325494523*13031*151*15151*989303989*131
*565*
33733*949*111*12121
48# = 14941*68786*348161843*36863*31313*14141*325494523*1458736378541*151*15151*959*131*565
*
1911191*979*3967693*989*111
49# = 454*14941*7894987*194484491*3242423*36863*31313*96199399169*325494523*13031*959*131
*
74988947*535*101*97679*949*111
50# = 35953*454*14941*7894987*194484491*3242423*36863*31313*96199399169*325494523*79597*131
*
74988947*535*101*97679*949*111
51# = 835538*35953*54040604045*14941*7894987*194484491*1279756579721*36863*325494523*959
*
7207027*1131311*101*949*93439*111
52# = 18881*835538*34586068543*7947497*7894987*33233*1174711*36863*31313*972131279*151*10001
*
131*7207027*1131311*515*101*76067*111
53# = 51815*18881*342969243*1270721*1426241*14941*160434061*348161843*36863*31313*7973797*151
*
15151*10001*131*1681861*2214122*30503*161
54# = 58985*3574753*1593223951*173535371*3203023*14941*137303666303731*194484491*191*600979006
*
325494523*13031*151*1131311*93439*111
55# = 55255*66766*16855255861*9438349*342969243*3203023*1426241*1420241*3626006263*36863*31313
*96199399169*1279721*
13031*15151*131*949
56# = 321123*9616169*3206023*18881*741373147*6474746*13493439431*33233*1279756579721*36863
*13031
*151*3296923*9409049*10001*7207027*515*101
57# = 99799*761282167*1240028200421*9515159*372296692273*4524687864254*5607065*14941*36863
*
31313*97401910479*151*15151*9050509*3967693
58# = 11111*99799*761282167*18727572781*58985*7166617*9438349*342969243*3203023*68786
*
3626006263*36863*31313*1279721*151*15151*92929*76067*767
59# = 11911*11111*99799*34400100443*198959891*7166617*1056776501*342969243*35953*14941
*561319291
3165*1160682860611*1279756579721*36863*2262622
60# = 37373*11911*11111*3194913*7444088804447*198959891*3206023*56165*7163223617*3203023
*14262
41*160434061*194484491*36863*31313*4125214*131*9182819
61# = 3113*83738*11911*14736863741*1498810188941*9694969*18727572781*3206023*992696299
*
741373147*3203023*587515785*33233*181*15151*10001*1911191*72427
62# = 3223*31413*719909917*11911*388294606492883*172606271*9694969*18727572781*7166617*9438349
*1
202021*16350105361*201686102*7973797*151*15151*70007*565
63# = 601106*32523*397606793*14444344441*7216127*55555*103707301*34400100443*18727572781*910464019
*741373147*35953*9503480843
059*1306031*191*31313*151*767
64# = 7842487*601106*3255523*788151887*14444344441*1197800087911*741751313157147*9694969*18727572781
*3206023*741373147*181*
14141*15151*515*1911191*99499*767

Regarding the multiple solutions:

The smallest example is for #39. There are at least 41 solutions.Like:

325494523*9958085808599*536011484110635*76867*131*73337*1131311*99499*6667666
or
325494523*85029792058*70807*151*76867*989303989*131*7207027*1131311*949*76067*555
or
325494523*7973797*14465856441*1491855581941*15151*8582858*73337*535*101*33733*989

***

Jan wrote:

There are only two solutions of the required type until & including 89#, the one you gave and 19#=33*494*595.
I see no obvious reason why not more might exist.

If one first constructs the potential palindromic factors (the time-consuming step), finding the split that equals p# is easy since the distribution of the primes p[i] in p# over the found primorials is not uniform.

For instance for 29# we get [9,8,3,13,19,10,8,9,3,1] as the frequencies for the primes from 2 to 29 on a total of 25 different primorials. The primorial 7337=11*23*29 is the only one containing 29. Crossing out the primorials having  factor 11 or 23 leaves 5 primorials, with 494=2*13*19 the only one containing 13. Crossing out these primes only leaves 595=5*7*17, therefore showing that no solution exists, since the prime 3 can not be reached. The point is that there seems to be a relative large probability for this process to end before a solution is found.

***

Emmanuel wrote:

I examined all primorials up to  101#.I found just one more solution : 9699690 = 19# = 33·494·595.

If we remove the condition that the palindromes should have at least two digits, then there are (besides the trivial cases  2#, 3#, 5#, 7#, 11#) a few more other solutions :

30030 = 13# = 2·3·5·1001
9699390 = 19# = 2·3·5·323·1001
223092870 = 23# = 5·161·323·858
6469693230 = 29# = 3·494·595·7337
And no more solutions up to 101#

***

Hakan wrote:

510510 is the smallest example. An other example: 19# = 9699690 = 33 x 494 x 595

There are no other example for the first 30 primorials.

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