[ March 1999 ] An original puzzle involving primes
Enoch Haga
Find more palindromes that are the concatenation of the nth prime with the sum of the primes smaller or equal to this nth prime. |
Enoch Haga himself discovered two nice solutions for this hard problem.
At 73, the 21st prime, the sum of the primes ⩽ 73 is 712
from which the palindrome 21712 is formed.
At 4177, the 574th prime, the sum of the primes ⩽ 4177 is 1111475
from which the palindrome 5741111475 is formed.
Apparently not easy to find, Enoch dares to challenge you to find more solutions !
"I have now checked to 199909, the 17978 th prime, and found nothing else to con-cat-enate!
Perhaps there are no more, but then I shall offer a prize of $5.95
(the sum of 21 and 574 divided by 100 -- just because it forms a palindrome)
to anyone discovering the next one in sequence (or who proves that it is impossible)."
[ August 15, 2002 ]
Jean Claude Rosa distinguished more cases that could be examined.
Let P be the prime number, N its rank number,
S the sum of the prime numbers ⩽ P,
& the concatenation operation and
PP the result that must be palindromic.
JCR proposes the following six 'equations' to solve !
N & S = PP (Enoch's puzzle)
S & N = PP
N & P = PP
P & N = PP
P & S = PP
S & P = PP
|
By varying P from 2 up to 1175497783 JCR obtained the following results :
| P | N | S | PP |
N & S | 73 4177 | 21 574 | 712 1111475 | 21712 5741111475 |
S & N | ? | ? | ? | ? |
N & P | 17 183661 61241363 | 7 16638 3631421 | | 717 16638183661 363142161241363 |
P & N |
491 1823 6883 757063 9642461 329147719 |
94 281 886 60757 642469 17741923 |
|
49194 1823281 (prime curios!) 6883886 75706360757 9642461642469 32914771917741923 |
P & S | 2 7 | 1 4 | 2 17 | 22 717 |
S & P | 2 | 1 | 2 | 22 |
|