Problems & Puzzles: Puzzles

Puzzle 1163� Palindromic primes that are semiprimes when turned upside down. G. L. Honaker, Jr.. sent the following nice puzzle. 191, 10601, 16061, ... These are palindromic primes that are semiprimes when turned upside down. For example, when turned upside down (rotated 180 degrees), you get 161 = 7 x 23 10901 = 11 x 991 19091 = 17 x 1123 https://t5k.org/glossary/page.php?sort=UpsideDown Q. Find more �

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From Feb 16-23, 2023, contributions came from Michael Branicky, J-M Rebert, Giorgos Kalogeropoulos, Jeff Heleen, Alessandro Casini, Paul Cleary, Gennady Gusev, Emmanuel Vantieghem, Oscar Volpatti

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Michael wrote:

These are quite common.� The first ten are

�

191, 10601, 16061, 19891, 1196911, 1600061, 1616161, 1660661, 1909091, 1998991
�

Attached is a file with�12247 such values, which are all those with <= 20 digits.

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J-M Rebert wrote:

I found 49815 numbers < 10^19. See attached file.

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Giorgos:

I am sending you in a txt�file the first 49815 such numbers

�

Here is also a 500-digit palindromic prime with this property

199688109060998618860168009106091608696866908100010901116089196819198668618699096660080181911
600880906609601891698988889168609898618909198916089190900686661601061988896900881991086966196
899809668000900806188090998168660868889181619116669919666980919091908966691996661191618198886
806686189909088160800900086690899869166968019918800969888916010616668600909198061989190981689
890686198888989619810690660908800611918108006669099681686689191869198061110901000180966869680
6190601900861068816899060901886991
�

�

when turned upside down (rotated 180 degrees), you get:
�

�

1669881060906689188901980061090619089698996081000106011190861698161689989189660699900801816119
0088060990690186196868888619890686891860616861908616060098999190109168886960088166108969916986
6806998000600809188060668198990898886181916119996616999680616061608699961669991161918168889809
9891866060881908006000899608668961996980166188006968886190109199989006061680916861606819868609
8916888868691681096099060880091161810800999606698198998616189616809111060100018069989698091609
01600891098819866090601889661 = 59 X�2830306882892693540511830612017998457116942510169671205408
2401663757457443892554235606810442695064510153712179994693031559136757443831554003234942474744
0664239138455711187935018796100287909842369492543744386543505422018472688443925627112910168950
1967145910168841864256981663879488304901467899799844077967626455371122028779942691537235400322
21898303221366386596078238444045527100286252011723408408458658984595962894932196723698288112138
80706456120440847488254219827273913586455781335929933739015079
�

�

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Jeff wrote:

I found these for pal-primes < 100,000,000:

�

191 --> 161 = 7 � 23
10601 --> 10901 = 11 � 991
16061 --> 19091 = 17 � 1123
19891 --> 16861 = 13 � 1297
1196911 --> 1169611 = 929 � 1259
1600061 --> 1900091 = 163 � 11657
1616161 --> 1919191 = 29 � 66179
1660661 --> 1990991 = 19 � 104789
1909091 --> 1606061 = 491 � 3271
1998991 --> 1668661 = 89 � 18749
9091909 --> 6061606 = 2 � 3030803
9801089 --> 6801086 = 2 � 3400543
9818189 --> 6818186 = 2 � 3409093
9888889 --> 6888886 = 2 � 3444443
9889889 --> 6886886 = 2 � 3443443
9908099 --> 6608066 = 2 � 3304033
9916199 --> 6619166 = 2 � 3309583
9918199 --> 6618166 = 2 � 3309083
9919199 --> 6616166 = 2 � 3308083
9981899 --> 6681866 = 2 � 3340933
9989899 --> 6686866 = 2 � 3343433

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Alessandro wrote:

There are many of them and it is quite easy to generate them. Therefore, just out of curiosity, I report some of them with certain additional properties chosen by me. In fact, there are some of these numbers such that one of the prime factor of the upside-down number satisfies the puzzle's property too.�Hence, in a certain sense these ones satisfy the property doubly. For example:
10668886601 when turned upside down has 191 in its prime factorization
1816869619169686181�when turned upside down has 10601�in its prime factorization
1066199901099916601 when turned upside down has 16061�in its prime factorization
Another interesting case is when the upside down number is a semiprime composed of no-trivial palprimes, as for 19166966191 (16199699161 = 101 � 160393061).
Eventually, 10819891801 when placed upside down is divisible by the prime 1163, the puzzle's number.
�

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Paul wrote:

There seems to be quite a lot of other solutions, I have linked a file containing over 3000 with the prime palindrome going up to�999991101199999.

�

If you would like more, let me know and I'll run my program a bit longer.

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Gennady wrote:

Here are some of the first results:

� � � � � � � � � � � �191 -> 161 = 7 * 23

� � � � � � � � � � 10601 -> 10901 = 11 * 991

� � � � � � � � � � 16061 -> 19091 = 17 * 1123

� � � � � � � � � � 19891 -> 16861 = 13 * 1297

� � � � � � � � �1196911 -> 1169611 = 929 * 1259

� � � � � � � � �1600061 -> 1900091 = 163 * 11657

� � � � � � � � �1616161 -> 1919191 = 29 * 66179

� � � � � � � � �1660661 -> 1990991 = 19 * 104789

� � � � � � � � �1909091 -> 1606061 = 491 * 3271

� � � � � � � � �1998991 -> 1668661 = 89 * 18749

� � � � � � � � �9091909 -> 6061606 = 2 * 3030803

� � � � � � � � �9801089 -> 6801086 = 2 * 3400543

� � � � � � � � �9818189 -> 6818186 = 2 * 3409093

� � � � � � � � �9888889 -> 6888886 = 2 * 3444443

� � � � � � � � �9889889 -> 6886886 = 2 * 3443443

� � � � � � � � �9908099 -> 6608066 = 2 * 3304033

� � � � � � � � �9916199 -> 6619166 = 2 * 3309583

� � � � � � � � �9918199 -> 6618166 = 2 * 3309083

� � � � � � � � �9919199 -> 6616166 = 2 * 3308083

� � � � � � � � �9981899 -> 6681866 = 2 * 3340933

� � � � � � � � �9989899 -> 6686866 = 2 * 3343433

� � � � � � � 100161001 -> 100191001 = 223 * 449287

� � � � � � � 100999001 -> 100666001 = 71 * 1417831

� � � � � � � 101898101 -> 101868101 = 19 * 5361479

� � � � � � � 101999101 -> 101666101 = 41 * 2479661

� � � � � � � 106909601 -> 109606901 = 1597 * 68633

� � � � � � � 108919801 -> 108616801 = 2777 * 39113

� � � � � � � 109909901 -> 106606601 = 41 * 2600161

� � � � � � � 110999011 -> 110666011 = 41 * 2699171

� � � � � � � 111191111 -> 111161111 = 1721 * 64591

� � � � � � � 111686111 -> 111989111 = 13 * 8614547

� � � � � � � 116000611 -> 119000911 = 67 * 1776133

� � � � � � � 116919611 -> 119616911 = 521 * 229591

� � � � � � � 118686811 -> 118989811 = 31 * 3838381

� � � � � � � 118909811 -> 118606811 = 499 * 237689

� � � � � � � 119868911 -> 116898611 = 83 * 1408417

� � � � � � � 160696061 -> 190969091 = 3671 * 52021

� � � � � � � 161969161 -> 191696191 = 23 * 8334617

� � � � � � � 166888661 -> 199888991 = 59 * 3387949

� � � � � � � 168191861 -> 198161891 = 109 * 1817999

� � � � � � � 168818861 -> 198818891 = 1511 * 131581

� � � � � � � 169686961 -> 196989691 = 491 * 401201

� � � � � � � 186101681 -> 189101981 = 151 * 1252331

� � � � � � � 188616881 -> 188919881 = 853 * 221477

� � � � � � � 189080981 -> 186080681 = 1559 * 119359

� � � � � � � 191090191 -> 161060161 = 67 * 2403883

� � � � � � � 191868191 -> 161898161 = 179 * 904459

� � � � � � � 191969191 -> 161696161 = 11 * 14699651

� � � � � � � 908808809 -> 608808806 = 2 * 304404403

� � � � � � � 908888809 -> 608888806 = 2 * 304444403

� � � � � � � 980888089 -> 680888086 = 2 * 340444043

� � � � � � � 981919189 -> 681616186 = 2 * 340808093

� � � � � � � 986000689 -> 689000986 = 2 * 344500493

� � � � � � � 989868989 -> 686898686 = 2 * 343449343

� � � � � � � 996181699 -> 669181966 = 2 * 334590983

� � � � � �10000900001 -> 10000600001 = 7 * 1428657143

� � � � � �10001610001 -> 10001910001 = 42773 * 233837

� � � � � �10016961001 -> 10019691001 = 77023 * 130087

� � � � � �10061916001 -> 10091619001 = 65111 * 154991

� � � � � �10100600101 -> 10100900101 = 67 * 150759703

� � � � � �10189898101 -> 10186868101 = 6581 * 1547921

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Emmanuel wrote:

It was not difficult to find more of such numbers (which I give the name "turnable numbers" for brevity).

Here are the next hundred�:

19891, 1196911, 1600061, 1616161, 1660661, 1909091, 1998991, 9091909, \
9801089, 9818189, 9888889, 9889889, 9908099, 9916199, 9918199, \
9919199, 9981899, 9989899, 100161001, 100999001, 101898101, \
101999101, 106909601, 108919801, 109909901, 110999011, 111191111, \
111686111, 116000611, 116919611, 118686811, 118909811, 119868911, \
160696061, 161969161, 166888661, 168191861, 168818861, 169686961, \
186101681, 188616881, 189080981, 191090191, 191868191, 191969191, \
908808809, 908888809, 980888089, 981919189, 986000689, 989868989, \
996181699, 10000900001, 10001610001, 10016961001, 10061916001, \
10100600101, 10189898101, 10606660601, 10611811601, 10619191601, \
10668886601, 10688988601, 10801610801, 10819891801, 10861016801, \
10888688801, 10890109801, 10898689801, 10901610901, 10960006901, \
10980808901, 10989898901, 11006660011, 11006960011, 11066666011, \
11089898011, 11089998011, 11098689011, 11106160111, 11116961111, \
11119991111, 11180608111, 11199199111, 11606960611, 11616161611, \
11619891611, 11668886611, 11689898611, 11690109611, 11699999611, \
11801610811, 11809890811, 11809990811, 11816961811, 11861116811, \
11869896811, 11869996811, 11989098911, 11991819911

�

(In annex I send you a list of 3026 elements, all < 10^15)

�

I spent the rest of the week looking for cases in which the two factors were palprimes too.

Here are the first 34 :

9888889, 9889889, 9908099, 9989899, 908808809, 908888809, 980888089, \
19166966191, 19966966991, 90998089909, 90999899909, 98089998089, \
98808880889, 98888988889, 99088988099, 9008089808009, 9008888888009, \
9080089800809, 9080890980809, 9098989898909, 9890988890989, \
9980890980899, 9980980890899, 9980990990899, 9988098908899, \
116968696869611, 181601808106181, 199669999966991, 900888080888009, \
900890888098009, 900900808009009, 900998888899009, 908900000009809, \
908980000089809, 908998989899809, 980089999980089, 988889909988889, \
988909999909889, 989080999080989, 989090808090989, 989090888090989, \
989909000909989, 998908000809899
�

One of these,�181601808106181� turns into 181901808109181 = 101*1801008001081,

both factors being "turnable".

�

I tried to find an example of a "turnable palprime" which turns into a semiprime with both factors "turnable palprimes" which turn into semiprimes.

But that was undoubtedly much too ambitious : I found none below� 10^23.�

�

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Oscar wrote:

Next solution is palprime 19891, producing semiprime 16861 = 13 * 1297.
There are 49815 solutions below 10^19. See attached file P1163small.txt.
For 209 such solutions, both factors of resulting semiprime are palprimes too.
The first few examples share a very simple form:
9888889� -> 6888886 = 2 * 3444443
9889889� -> 6886886 = 2 * 3443443
9908099� -> 6608066 = 2 * 3304033
9989899� -> 6686866 = 2 * 3343433
Palprime p only contains digits 9, 8, and (eventually) 0; it begins and ends with 9, digit 8 is used at least once.
Turning p upside down, every 9 becomes a 6, so the resulting number admits 2 as proper factor, with an odd cofactor only containing digits 3, 4, and eventually 0 (with the same palindrome�pattern of starting prime).
I searched for more solutions of the special form "9...90...08...80...09...9", where equal digits are grouped into five blocks, until I found a titanic palprime.�
See attached file P1163titanic90809.txt, where every solution is coded using three parameters:�
total length, number of starting�nines, number of starting zeros.
9888889 is coded as (7,1,0)

9908099 is coded as (7,2,1)

908888809 is coded as (9,1,1)

and so on; the first such titanic solution is coded as (1013,296,180).


Finally, two very nice solutions:
181601808106181 -> 181901808109181 = 101 * 1801008001081
In these cases, both factors of resulting semiprime are tetradic primes.

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Oscar, Giorgos and J-M Rebert explored up to integers with 20 digits and the three of them report the same number of solutions: 49,815.

I have chosen arbitrarily the file from Volpatti to add it here.

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