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Mirror, mirror (Posted on 2012-05-04) Difficulty: 3 of 5
What palindromic prime number of n digits reads the same upside down or when viewed in a mirror?

Some examples for n=2 or 3:
11, 101, 181.

How far can you go?

Rem: Only digits 0,l and 8 qualify.

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution computer solution | Comment 3 of 4 |

    4   kill "mirrmir3.txt"
    5   open "mirrmir3.txt" for output as #2
   10   No2="1"
   20   gosub *AddOn
   25   close #2
   30   end
   40   '
   50   *AddOn
   60   No2=No2+"0"
   70   gosub *TestNo2
   80   if len(No2)<13 then gosub *AddOn
   90   No2=left(No2,len(No2)-1)+"1"
  100   gosub *TestNo2
  110   if len(No2)<13 then gosub *AddOn
  120   No2=left(No2,len(No2)-1)+"8"
  130   gosub *TestNo2
  140   if len(No2)<13 then gosub *AddOn
  150   No2=left(No2,len(No2)-1)
  160   return
  170 
  180   *TestNo2
  190   Tst=No2
  200   for I=len(No2)-1 to 1 step -1
  210     Tst=Tst+mid(No2,I,1)
  220   next I
  230   if fnPrime(val(Tst)) then print #2,spc(27-len(Tst))+Tst
  240   return
  300 
10000   fnOddfact(N)
10010   local K=0,P
10030   while N@2=0
10040     N=N\2
10050     K=K+1
10060   wend
10070   P=pack(N,K)
10080   return(P)
10090   '
10100   fnPrime(N)
10110   local I,X,J,Y,Q,K,T,Ans
10120   if N@2=0 then Ans=0:goto *EndPrime
10125   O=fnOddfact(N-1)
10130   Q=member(O,1)
10140   K=member(O,2)
10150   I=0
10160   repeat
10170     repeat
10180       X=fnLrand(N)
10190     until X>1
10200     J=0
10210     Y=modpow(X,Q,N)
10220     loop
10230       if or{and{J=0,Y=1},Y=N-1} then goto *ProbPrime
10240       if and{J>0,Y=1} then goto *NotPrime
10250       J=J+1
10260       if J=K then goto *NotPrime
10270       Y=(Y*Y)@N
10280     endloop
10290    *ProbPrime
10300     I=I+1
10310   until I>50
10320   Ans=1
10330   goto *EndPrime
10340   *NotPrime
10350   Ans=0
10360   *EndPrime
10370   return(Ans)
10380   '
10400   fnLrand(N)
10410   local R
10415   N=int(N)
10420   R=(int(rnd*10^(alen(N)+2)))@N
10430   return(R)
10440   '

finds 36,499 odd-length such primes up through length 25. The only even-length palindromic prime is 11, as all even-length palindromes are divisible by 11. So, adding in that one prime, makes 36,500 such primes through length 25.

The program uses a probabilistic prime test that makes it astronomically unlikely that a composite will be declared prime.

The list was created in lexicographic order and then the file was sorted into numeric order. I'll try to get up through the 17-digit ones to list here:

                        101
                        181
                      18181
                    1008001
                    1180811
                    1880881
                    1881881
                  100111001
                  100888001
                  108101801
                  110111011
                  111010111
                  111181111
                  118818811
                  180101081
                  181111181
                  181888181
                  188010881
                  188888881
                10008180001
                10081818001
                10101810101
                10180008101
                10180808101
                10181018101
                10810001801
                10880108801
                11008080011
                11088888011
                11100800111
                11188888111
                11800000811
                11880108811
                11881018811
                11881818811
                11888088811
                18001010081
                18018881081
                18081818081
                18110001181
                18111111181
                18118881181
                18180808181
                18810101881
                18818181881
              1000008000001
              1000180810001
              1000188810001
              1008008008001
              1008100018001
              1008800088001
              1010088800101
              1010181810101
              1010810180101
              1018101018101
              1018808088101
              1080008000801
              1080081800801
              1081088801801
              1081810181801
              1088800088801
              1100011100011
              1100018100011
              1100080800011
              1100101010011
              1100118110011
              1100881880011
              1101008001011
              1101010101011
              1101081801011
              1101181811011
              1101188811011
              1101880881011
              1108088808011
              1108881888011
              1110110110111
              1110111110111
              1110811180111
              1110880880111
              1111008001111
              1111118111111
              1111188811111
              1118011108111
              1118888888111
              1180000000811
              1180010100811
              1180818180811
              1181018101811
              1188811188811
              1800010100081
              1800018100081
              1800818180081
              1801008001081
              1801088801081
              1801808081081
              1801880881081
              1808110118081
              1810008000181
              1810100010181
              1811018101181
              1880000000881
              1880108010881
              1880808080881
              1881008001881
              1881100011881
              1881800081881
              1888081808881
            100008101800001
            100018080810001
            100081101180001
            100100888001001
            100108818801001
            100110101011001
            100110808011001
            100118808811001
            100181111181001
            100181181181001
            100188080881001
            100800818008001
            100801000108001
            100808181808001
            100810080018001
            100818080818001
            100881080188001
            100888111888001
            100888818888001
            101000010000101
            101010080010101
            101011000110101
            101011888110101
            101018000810101
            101080888080101
            101081888180101
            101110000011101
            101110818011101
            101118010811101
            101188101881101
            101800080008101
            101800101008101
            101800818008101
            101801181108101
            101801808108101
            101810181018101
            101811010118101
            101811808118101
            101888010888101
            108000181000801
            108001111100801
            108008000800801
            108008808800801
            108010080010801
            108081181180801
            108100080001801
            108100101001801
            108108010801801
            108108101801801
            108110000011801
            108110888011801
            108188808881801
            108800181008801
            108800808008801
            108881010188801
            108888010888801
            110000808000011
            110000888000011
            110011101110011
            110018010810011
            110080111080011
            110081181180011
            110081808180011
            110108808801011
            110118818811011
            110180111081011
            110188888881011
            110800818008011
            110800888008011
            110808080808011
            110808808808011
            110810888018011
            110880818088011
            110881000188011
            111001888100111
            111008010800111
            111008808800111
            111008888800111
            111010888010111
            111011808110111
            111011888110111
            111088000880111
            111088080880111
            111100111001111
            111101888101111
            111111181111111
            111180181081111
            111181080181111
            111188111881111
            111188181881111
            111188888881111
            111800181008111
            111801101108111
            111810181018111
            111810808018111
            111880101088111
            111881010188111
            111888181888111
            118000000000811
            118000101000811
            118010000010811
            118011181110811
            118011818110811
            118018080810811
            118018101810811
            118018818810811
            118088181880811
            118100111001811
            118101101101811
            118111080111811
            118118111811811
            118118181811811
            118800101008811
            118808888808811
            118811808118811
            118818010818811
            118818101818811
            118818181818811
            118881080188811
            118881808188811
            180000010000081
            180000808000081
            180001101100081
            180001888100081
            180008808800081
            180011808110081
            180088888880081
            180100080001081
            180100101001081
            180101808101081
            180108080801081
            180108808801081
            180118000811081
            180180080081081
            180180101081081
            180800111008081
            180801101108081
            180810080018081
            180811000118081
            180880888088081
            181000888000181
            181011808110181
            181081888180181
            181110101011181
            181110808011181
            181111818111181
            181188818881181
            181800080008181
            181801818108181
            181811111118181
            181818818818181
            188001101100881
            188010080010881
            188010181010881
            188011111110881
            188011818110881
            188018888810881
            188081000180881
            188081888180881
            188100101001881
            188108111801881
            188110101011881
            188110111011881
            188118818811881
            188180181081881
            188181101181881
            188188111881881
            188808080808881
            188818111818881
            188818888818881
            188881111188881
            188881818188881
            188888888888881
          10000010101000001
          10000181018100001
          10000188188100001
          10000800800800001
          10000801010800001
          10000818181800001
          10000888088800001
          10000888888800001
          10001808880810001
          10008000800080001
          10008008080080001
          10008011011080001
          10008101010180001
          10008810101880001
          10008811811880001
          10008880008880001
          10010008080001001
          10010088888001001
          10010100800101001
          10010801010801001
          10011001810011001
          10011010001011001
          10011080108011001
          10011088088011001
          10011118881111001
          10011808080811001
          10011811011811001
          10011888888811001
          10018180008181001
          10018811111881001
          10018818881881001
          10018888888881001
          10080000100008001
          10080001010008001
          10080001810008001
          10080100000108001
          10080180808108001
          10080810101808001
          10080818181808001
          10081000000018001
          10081001110018001
          10081081018018001
          10081108080118001
          10081810001818001
          10088080808088001
          10088081018088001
          10088180008188001
          10088801110888001
          10088811111888001
          10088818081888001
          10100081018000101
          10100081118000101
          10100110001100101
          10100110801100101
          10101001810010101
          10101180808110101
          10101888188810101
          10108001010080101
          10108018881080101
          10108108080180101
          10108800800880101
          10108880108880101
          10110001810001101
          10110010001001101
          10110088888001101
          10110101110101101
          10110108880101101
          10110811811801101
          10110880008801101
          10111000000011101
          10111101110111101
          10111110101111101
          10111880008811101
          10118001110081101
          10118010801081101
          10180080108008101
          10180080808008101
          10180088088008101
          10180808180808101
          10180810001808101
          10181011811018101
          10188001010088101
          10188011011088101
          10188018181088101
          10188101110188101
          10188118081188101
          10188180808188101
          10188810101888101
          10188818081888101
          10800008180000801
          10800110001100801
          10800118081100801
          10800181018100801
          10801000800010801
          10801001110010801
          10801011811010801
          10801100000110801
          10801180808110801
          10801800100810801
          10801801010810801
          10801811011810801
          10801888888810801
          10808008880080801
          10808180108180801
          10808181118180801
          10808188088180801
          10808800000880801
          10808810101880801
          10808880808880801
          10808881018880801
          10810008880001801
          10810011811001801
          10810081818001801
          10810100100101801
          10810101810101801
          10810118181101801
          10810800100801801
          10811001110011801
          10811008880011801
          10811010801011801
          10811108180111801
          10811818081811801
          10811881018811801
          10818000800081801
          10818008080081801
          10818088088081801
          10818808880881801
          10818880808881801
          10880010101008801
          10880011011008801
          10880080808008801
          10880101010108801
          10880118181108801
          10880801110808801
          10880811811808801
          10881018181018801
          10881088888018801
          10881100000118801
          10881110001118801
          10881880008818801
          10881881018818801
          10888010101088801
          10888108880188801
          11000081118000011
          11000188888100011
          11000800000800011
          11000800800800011
          11000810101800011
          11000888088800011
          11000888188800011
          11001010801010011
          11001100800110011
          11001111811110011
          11001118081110011
          11001801810810011
          11008000000080011
          11008100100180011
          11008110101180011
          11008810801880011
          11008880008880011
          11010000800001011
          11010001010001011
          11010081118001011
          11010180108101011
          11010880808801011
          11011118081111011
          11011801010811011
          11011818181811011
          11011881118811011
          11018008880081011
          11018188088181011
          11018881818881011
          11080080808008011
          11080101110108011
          11080180008108011
          11081080008018011
          11081088188018011
          11081880108818011
          11081888188818011
          11088008080088011
          11088011111088011
          11088110101188011
          11088188088188011
          11088880808888011
          11100088088000111
          11100110001100111
          11100188188100111
          11101000800010111
          11101008880010111
          11101018881010111
          11101088088010111
          11108818081880111
          11110118081101111
          11110188188101111
          11110888888801111
          11111011811011111
          11111081118011111
          11111118881111111
          11111801110811111
          11111808180811111
          11118001010081111
          11118008180081111
          11118010001081111
          11180000100008111
          11180110801108111
          11180188088108111
          11181000800018111
          11181010801018111
          11181101110118111
          11181118881118111
          11181888188818111
          11188000800088111
          11188080108088111
          11188101110188111
          11188108880188111
          11800081818000811
          11800088088000811
          11800801010800811
          11800810001800811
          11801001110010811
          11801010101010811
          11801800000810811
          11801808180810811
          11801810001810811
          11808001010080811
          11808018181080811
          11808081118080811
          11810000000001811
          11810010801001811
          11810080808001811
          11810081018001811
          11810118881101811
          11810801110801811
          11810818881801811
          11811010801011811
          11811018181011811
          11811111111111811
          11811118881111811
          11811811811811811
          11811818181811811
          11818001110081811
          11818011811081811
          11818088188081811
          11818111811181811
          11818801010881811
          11818801810881811
          11818808080881811
          11818808180881811
          11818818181881811
          11818818881881811
          11880101110108811
          11880108880108811
          11880801110808811
          11881008180018811
          11881011111018811
          11881810001818811
          11888008880088811
          11888081118088811
          11888101010188811
          11888800800888811
          11888811811888811
          11888818881888811
          18000000100000081
          18000001010000081
          18000818181800081
          18000888188800081
          18001008180010081
          18001008880010081
          18001010001010081
          18001111811110081
          18001188188110081
          18001188888110081
          18001810801810081
          18008118181180081
          18008180108180081
          18008181118180081
          18008880808880081
          18010001110001081
          18010008880001081
          18010010101001081
          18010011811001081
          18010088088001081
          18010088188001081
          18010101110101081
          18010101810101081
          18010181018101081
          18011008080011081
          18011011811011081
          18011081118011081
          18011180108111081
          18011181118111081
          18011811111811081
          18018008080081081
          18018008180081081
          18018100000181081
          18018111811181081
          18018118081181081
          18018181818181081
          18018188188181081
          18018800100881081
          18018808080881081
          18018880808881081
          18080800000808081
          18080810101808081
          18081081118018081
          18081180108118081
          18081810801818081
          18081811811818081
          18081888188818081
          18088011811088081
          18088018881088081
          18088111011188081
          18088181018188081
          18088800100888081
          18088808080888081
          18088880008888081
          18100011811000181
          18100088088000181
          18100111811100181
          18100118181100181
          18100818181800181
          18101808180810181
          18101880108810181
          18101881018810181
          18108000800080181
          18108008080080181
          18108011011080181
          18110080008001181
          18110111011101181
          18110188088101181
          18110810101801181
          18111080108011181
          18111088088011181
          18111808880811181
          18111811811811181
          18111880008811181
          18118000000081181
          18118010001081181
          18118088188081181
          18118118181181181
          18118181118181181
          18180000800008181
          18180011011008181
          18180081118008181
          18180108880108181
          18180118881108181
          18180181818108181
          18180810101808181
          18180811811808181
          18180888188808181
          18181008180018181
          18181080808018181
          18188010801088181
          18188011011088181
          18188011811088181
          18188018081088181
          18800081118000881
          18800100800100881
          18800110001100881
          18800118881100881
          18800188888100881
          18800801010800881
          18800811111800881
          18800818881800881
          18801001810010881
          18801088088010881
          18808008880080881
          18808011111080881
          18808088188080881
          18808880108880881
          18808888088880881
          18810101110101881
          18810111811101881
          18810801810801881
          18810818081801881
          18810880808801881
          18811108080111881
          18811801810811881
          18818101810181881
          18818808880881881
          18880108880108881
          18880110101108881
          18880111011108881
          18880188888108881
          18880818181808881
          18880888088808881
          18881011011018881
          18881088888018881
          18888001810088881
          18888008180088881
          18888080008088881
          18888108880188881

  Posted by Charlie on 2012-05-04 13:02:36
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