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.

>
>
> János
>

You're welcome :-)

Peter

```

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