Problems & Puzzles: Puzzles

 Puzzle 659 Sandwiched palindromes Here we ask for palindrome integers P such that P+1 & P-1 are prime integers. The smallest example is 282. Q. Send your largest &/or nice P case.

Contributions came from Torbjörn Alm, W. Edwin Clark, Giovanni Resta,  Emmanuel Vantieghem, Hakan Summakoglu, Farideh Firoozbakht & Igor Schein.

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Alm wrote:

Here is my best solution. Solutions to (p-1,p+1 prime, p palimdrome
Solution P= 8999999993443999999998. Largest solution with 22 digits

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Clark wrote:

Here's a 402 digit palindrome n such that n-1 and n+1 are primes
(as determined by Maple's probabilistic primality test isprime):

8630060000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000011000000\
0000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000\
00000000000000000000000000000000600368

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Giovanni wrote:

For the present puzzle, I will limit my contribution to the smallest such palindrome with 1000 digits. It is equal to 2 10^999 + 230950059032*10^494 + 2... and this 900-digit palindrome

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 8
8 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 8
8 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 8
8 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

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Emmanuel wrote:

I examined many  P  of the form  A*10^k + R(A)  for several values of  A  and  k.  I found lots of beautifull  P  for which {P-1,P+1} is a twin pair.  My biggest hit isP = 22653*10^1435 + 35622 (1440 digits). The primality of  P-1  and  P+1  is certified by PRIMO.

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Hakan wrote:

I checked for <10^13 and found 14503 palindromes with these conditions.

Solutions:1-digit=0, 2-digit=0, 3-digit=3 , 4-digit=1 , 5-digit=12 , 6-digits=4 , 7-digit=36 , 8-digit=28 , 9-digit=220 , 10-digit=232 , 11-digit=1641 , 12-digit=1624 , 13-digit=10702 Only for 4-digit, there is 1 solution. (2112: 2111 and 2113 are primes)

I also checked for smallest nested sandwiched palindromes.

For i=1 ,Smallest solution is 282.

1:282 : (281 and 283 are primes)

For i=2 ,Smallest solution is 2182812.

1:2182812 : (2182811 and 2182813 are primes)

2:   828     : (827 and 829 are primes)

For i=3 ,Smallest solution is 2132428242312.

1: 2132428242312 : ( 2132428242311 and 2132428242313 are primes)

2:      2428242      : ( 2428241 and 2428243 are primes)

3:         282          : (281 and 283 are primes)

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Jahangeer Kholdi & Farideh Firoozbakht wrote:

P = 8(10^274-1)/9+10^81(10^112-1)/9 = 8(81).9(112).8(81) is a 274 - digit  palindrome
with the property that both numbers P-1 & P+1 are prime.

P = 8888888888888888888888888888888888888888888888888888888
8888888888888888888888888899999999999999999999999999999
9999999999999999999999999999999999999999999999999999999
9999999999999999999999999999888888888888888888888888888
888888888888888888888888888888888888888888888888888888

We also found three nice palindromes P1, P2 & P3 with the requested property.

P1 = 8111111118

P2 = 2*(10^31-1)/9 + 20*(10^29-1)/9
= 2.4(29).2
= 2444444444444444444444444444442

P3 = 8*(10^55-1)/9 - 8*10^27
= 8(27).0.9(27)
= 8888888888888888888888888880888888888888888888888888888

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Igor wrote:

Here's an example of a 384-digit number from that sequence:

899999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999987580085789999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999
999999998

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Emmanuel Vantieghem wrote (Nov. 2012)

About puzzle 659, this is the smallest solution with 2000 digits :
m = 10^1999+21036377363012 10^993+2
PRIMO needed about 11 hours to prove the primality of  m - 1  and  m+1.

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