Re: compound symmetrical primes

*To*: mathgroup at smc.vnet.net*Subject*: [mg66503] Re: compound symmetrical primes*From*: Peter Pein <petsie at dordos.net>*Date*: Wed, 17 May 2006 03:29:48 -0400 (EDT)*References*: <e4birv$1a5$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

János schrieb: > Let's say that a prime is symmetrical if prime==FromDigits[Reverse > [IntegerDigits[prime]]]. > > I would say a prime is compound if two contiguous distinct subsets of > its digits are summing up to the same number. For example 211 is a > compound prime because 2=1+1. Similarly 15877 is a compound prime > because 1+5+8=7+7. May be there is a better definition in the prime > literature. The two distinct subset has to cover all the digits. > > Here is a little program that that looks for symmetrical compound > primes up to an mx limit. > > In[14]:= > lst = Timing[First[ > Last[Reap[i = 1; > mx = 10^6; While[ > i <= mx, > pr = Prime[i]; > If[pr != FromDigits[ > Reverse[ > IntegerDigits[ > pr]]], i++; > Continue[]; ]; > prdig = > IntegerDigits[pr]; > prlen = Length[ > prdig]; j = 1; > While[j < prlen, > prLeft = Take[ > prdig, {1, j}]; > prRight = Take[ > prdig, {j + 1, > prlen}]; > If[Total[prLeft] != > Total[prRight], > j++; Continue[], > Sow[pr]; Break[ > ]]; ]; i++; ]; ]]]] > Out[14]= > {31.687534999999997*Second, > {11, 101, 16061, 31013, > 35053, 38083, 73037, > 74047, 91019, 94049, > 1120211, 1150511, 1160611, > 1180811, 1190911, 1250521, > 1280821, 1300031, 1360631, > 1390931, 1490941, 1520251, > 1550551, 1580851, 1600061, > 1630361, 1640461, 1660661, > 1670761, 1730371, 1820281, > 1880881, 1930391, 1970791, > 3140413, 3160613, 3260623, > 3310133, 3380833, 3400043, > 3460643, 3470743, 3590953, > 3670763, 3680863, 3970793, > 7100017, 7190917, 7250527, > 7300037, 7310137, 7540457, > 7600067, 7630367, 7690967, > 7750577, 7820287, 7850587, > 7930397, 7960697, 9110119, > 9200029, 9230329, 9280829, > 9320239, 9400049, 9440449, > 9470749, 9610169, 9620269, > 9650569, 9670769, 9700079, > 9770779, 9820289, 9980899}} > > Here are a few questions: > > Is there any compound symmetrical prime other than 11 whose length is > even ? No, they are divisible by 11. > > Is there any compound symmetrical prime where the middle digit is not > zero ? No, since the subsets of the digits have to be contigous, the middle digit has to be zero. > > Is there a much faster algorithm to find these numbers ? /I am > mostly procedural here :) because I could not find a functional check > for compoundness. / > I searched up to mx=10^8 and on my little iBook it took the whole night. This finds the "constructors" of all the compound symmetrical primes up to 10^(2*6-1): {t,mylst2}=Timing@Select[Range[10^6], Function[tst, Module[{pr=Join[#,Piecewise[{{#,Length[#]==1}},Rest@Reverse[#]]]&@ IntegerDigits[tst],n}, n=FromDigits[pr]; PrimeQ[n]&& MemberQ[Tr/@Through[{Take,Drop}[pr,#]]&/@Range[Length[pr]], {x_Integer,x_}] ]]]; t --> 57.375 Second (your original code needs 18.766 to run on my computer) Length[mylst2] --> 4462 > > Let's say a number is periodical if it is a repetition of a subset of > its digits . For example 11 and 232323 are periodic numbers. > > Is 11 the only periodical symmetrical compound prime ? Every other periodical number is divisible by Sum[10^(k*p),k=0..n-1], p=length of the period, and n the count of repetitions of the period. > > Thanks ahead, > > János > You're welcome :-) Peter