A-positive-integer-such-as-4334-is-a-palindrome-if-it-reads-the-same-forwards-or-backwards-What-is-the-only-prime-palindrome-with-an-even-number-of-digits-

Question Number 100733 by john santu last updated on 28/Jun/20

A positive integer such as 4334 is

a palindrome if it reads the same

forwards or backwards . What is

the only prime palindrome with an

even number of digits ?

Commented by Rasheed.Sindhi last updated on 28/Jun/20

11

A n y o t h e r p a l i n d r o m e m u s t

b e d i v i s i b l e b y 11 a n d h e n c e

c o m p o s i t e .

Commented by 1549442205 last updated on 28/Jun/20

Thank you, sir . It is correct perfectly .

Commented by Rasheed.Sindhi last updated on 28/Jun/20

1331 = 11^{3}

S o {i t}^{'} s n o t p r i m e .

Commented by 1549442205 last updated on 07/Jul/20

We prove the general clause following :

^{″} a arbirtary palindrome number is

always divisible by 11^{″} . Indeedly, a palindrome

is always expressed in form :

A = \overset{―}{a_{1} a_{2} \dots a_{n} a_{n} a_{n - 1} \dots . . a_{1} =} \overset{―}{a_{1} a_{2} \dots a_{n}} . 10^{n} + \overset{―}{a_{n} a_{n - 1} \dots a_{1}}

= a_{1} (10^{2 n - 1} + 1) + {10 a}_{2} (10^{2 n - 3} + 1) + \dots

+ 10^{n - 2} a_{n - 1} . (10^{3} + 1) + 10^{n - 1} a_{n} (10 + 1)

This number is divisible by 11 because

10^{2 k + 1} + 1 = (10 + 1) (10^{2 k} - 10^{2 k - 1} + 10^{2 k - 2} - \dots . + 1) ⋮ 11

Commented by john santu last updated on 28/Jun/20

yes . . right

yes . . right

Commented by john santu last updated on 28/Jun/20

only 11

only 11