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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?

$$\mathrm{A}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{such}\:\mathrm{as}\:\mathrm{4334}\:\mathrm{is} \\ $$$$\mathrm{a}\:\mathrm{palindrome}\:\mathrm{if}\:\mathrm{it}\:\mathrm{reads}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{forwards}\:\mathrm{or}\:\mathrm{backwards}.\:\mathrm{What}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{only}\:\mathrm{prime}\:\mathrm{palindrome}\:\mathrm{with}\:\mathrm{an} \\ $$$$\mathrm{even}\:\mathrm{number}\:\mathrm{of}\:\mathrm{digits}?\: \\ $$

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

11 Any other palindrome must be divisible by 11 and hence composite.

$$\mathrm{11} \\ $$$${Any}\:{other}\:{palindrome}\:{must} \\ $$$${be}\:{divisible}\:{by}\:\mathrm{11}\:{and}\:{hence} \\ $$$${composite}. \\ $$

Commented by 1549442205 last updated on 28/Jun/20

Thank you,sir.It is correct perfectly.

$$\mathrm{Thank}\:\mathrm{you},\mathrm{sir}.\mathrm{It}\:\mathrm{is}\:\mathrm{correct}\:\mathrm{perfectly}. \\ $$

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

1331=11^3 So it′s not prime.

$$\mathrm{1331}=\mathrm{11}^{\mathrm{3}} \\ $$$${So}\:{it}'{s}\:{not}\:{prime}. \\ $$

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=a_1 a_2 ...a_n a_n a_(n−1) .....a_1 =^(−) a_1 a_2 ...a_n ^(−) .10^n +a_n a_(n−1) ...a_1 ^(−) =a_1 (10^(2n−1) +1)+10a_2 (10^(2n−3) +1)+... +10^(n−2) a_(n−1) .(10^3 +1)+10^(n−1) a_n (10+1) This number is divisible by 11 because 10^(2k+1) +1=(10+1)(10^(2k) −10^(2k−1) +10^(2k−2) −....+1)⋮11

$$\mathrm{We}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{general}\:\mathrm{clause}\:\mathrm{following}: \\ $$$$''\mathrm{a}\:\mathrm{arbirtary}\:\mathrm{palindrome}\:\mathrm{number}\:\mathrm{is} \\ $$$$\mathrm{always}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{11}''.\mathrm{Indeedly},\mathrm{a}\:\mathrm{palindrome} \\ $$$$\mathrm{is}\:\mathrm{always}\:\mathrm{expressed}\:\mathrm{in}\:\mathrm{form}: \\ $$$$\boldsymbol{\mathrm{A}}=\overline {\boldsymbol{\mathrm{a}}_{\mathrm{1}} \boldsymbol{\mathrm{a}}_{\mathrm{2}} ...\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}−\mathrm{1}} .....\boldsymbol{\mathrm{a}}_{\mathrm{1}} =}\overline {\boldsymbol{\mathrm{a}}_{\mathrm{1}} \boldsymbol{\mathrm{a}}_{\mathrm{2}} ...\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} }.\mathrm{10}^{\boldsymbol{\mathrm{n}}} +\overline {\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} \boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}−\mathrm{1}} ...\boldsymbol{\mathrm{a}}_{\mathrm{1}} } \\ $$$$=\mathrm{a}_{\mathrm{1}} \left(\mathrm{10}^{\mathrm{2n}−\mathrm{1}} +\mathrm{1}\right)+\mathrm{10a}_{\mathrm{2}} \left(\mathrm{10}^{\mathrm{2n}−\mathrm{3}} +\mathrm{1}\right)+... \\ $$$$+\mathrm{10}^{\mathrm{n}−\mathrm{2}} \mathrm{a}_{\mathrm{n}−\mathrm{1}} .\left(\mathrm{10}^{\mathrm{3}} +\mathrm{1}\right)+\mathrm{10}^{\mathrm{n}−\mathrm{1}} \mathrm{a}_{\mathrm{n}} \left(\mathrm{10}+\mathrm{1}\right) \\ $$$$\boldsymbol{\mathrm{This}}\:\boldsymbol{\mathrm{number}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{divisible}}\:\boldsymbol{\mathrm{by}}\:\mathrm{11}\:\boldsymbol{\mathrm{because}} \\ $$$$\mathrm{10}^{\mathrm{2}\boldsymbol{\mathrm{k}}+\mathrm{1}} +\mathrm{1}=\left(\mathrm{10}+\mathrm{1}\right)\left(\mathrm{10}^{\mathrm{2}\boldsymbol{\mathrm{k}}} −\mathrm{10}^{\mathrm{2}\boldsymbol{\mathrm{k}}−\mathrm{1}} +\mathrm{10}^{\mathrm{2}\boldsymbol{\mathrm{k}}−\mathrm{2}} −....+\mathrm{1}\right)\vdots\mathrm{11} \\ $$

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

yes..right

$$\mathrm{yes}..\mathrm{right} \\ $$

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

only 11

$$\mathrm{only}\:\mathrm{11} \\ $$

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