World!Of
Numbers  WON plate
56 |

[ 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 734177 21574 7121111475 217125741111475 S & N ? ? ? ? N & P 1718366161241363 7166383631421 71716638183661363142161241363 P & N 491182368837570639642461329147719 942818866075764246917741923 491941823281 (prime curios!)688388675706360757964246164246932914771917741923 P & S 27 14 217 22717 S & P 2 1 2 22  A000056 Prime Curios! Prime Puzzle  Wikipedia 56 Le Nombre 56 Numberland 56 ```

```

[ TOP OF PAGE]

Patrick De Geest - Belgium - Short Bio - Some Pictures