compound symmetrical primes

• To: mathgroup at smc.vnet.net
• Subject: [mg66482] compound symmetrical primes
• From: János <janos.lobb at yale.edu>
• Date: Mon, 15 May 2006 23:49:16 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

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 ?

Is there any compound symmetrical prime where the middle digit is not
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.

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 ?