2 characteristics of the row (1, 11, 111, 1 111, ...)

[Function]

Hello everyone

 

I have 2 questions:

 

First of all, I'd like to know if these (notations) are correct:

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)\equiv (1,0,1,0,1,0,\cdots )\bmod11[/math]

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)^2\equiv (1,0,1,0,1,0,\cdots )\bmod11[/math]

 

Secondly and finally, I'd like to know if it's possible to prove that the row

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)=(1,11,111,1111,11111,111111,\cdots)[/math] has no prime numbers in it (11 excluded)?

 

Thank you!

 

-Function

 

P.S. Could the title please be changed to "2 characteristics of the row (1, 11, 111, 1 111, ...)"? Thanks.

Edited January 11, 2014 by Function

[imatfaal]

Title amended

Hello everyone

 

I have 2 questions:

 

First of all, I'd like to know if these (notations) are correct:

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)\equiv (1,0,1,0,1,0,\cdots )\bmod11[/math]

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)^2\equiv (1,0,1,0,1,0,\cdots )\bmod11[/math]

 

Secondly and finally, I'd like to know if it's possible to prove that the row

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)=(1,11,111,1111,11111,111111,\cdots)[/math] has no prime numbers in it (11 excluded)?

 

Thank you!

 

-Function

 

P.S. Could the title please be changed to "2 characteristics of the row (1, 11, 111, 1 111, ...)"? Thanks.

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)=(1,11,111,1111,11111,111111,\cdots)[/math] has no prime numbers in it (11 excluded)?

 

I am not quite sure of your notation but it looks fine to me - but better check as my maths is mostly self-taught . If you are asking if it is possible to prove that no number of the form 1....1 (ie all digits are 1) is prime other than eleven; then I am afraid that the answer is - No you cannot prove it.

 

It is trivially disproved by counter-example 1,111,111,111,111,111,111 - ie 19 ones in a row. This number is prime - as are 23 ones in a row and a few (possibly infinite) more

 

https://oeis.org/A004023

[Function]

Title amended

 

[math]\left(u_n=\sum_{i=0}^{+\infty}{10^i}\right)=(1,11,111,1111,11111,111111,\cdots)[/math] has no prime numbers in it (11 excluded)?

 

I am not quite sure of your notation but it looks fine to me - but better check as my maths is mostly self-taught . If you are asking if it is possible to prove that no number of the form 1....1 (ie all digits are 1) is prime other than eleven; then I am afraid that the answer is - No you cannot prove it.

 

It is trivially disproved by counter-example 1,111,111,111,111,111,111 - ie 19 ones in a row. This number is prime - as are 23 ones in a row and a few (possibly infinite) more

 

https://oeis.org/A004023

 

1. Well, I didn't know if 'my' notation is correct, for this part of maths is self-taught to me too

2. Thanks