Exponents of ec primes ending with digit 7
up vote
0
down vote
favorite
A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.
In this question ec-numbers are introduced, obttained by the concatenation of two consecutive Mersenne numbers (12763=ec(7) for example). The values of $k$ for which $ec(k)$ is prime are: $[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]$
$7, 67, 360787$ are all the $k's$ leading to a prime ending with digit $7$. $7,67,360787$ are all congruent to $1$ $pmod 6$. My question is: does a k exist such that k ends with digit $7$, $ec(k)$ is prime and $k$ is NOT congruent to 1 $pmod 6$? Or that can be ruled out? In other words, if $k$ is congruent to 7 mod 10 and $ec(k)$ is prime, then necessarly $k$ must be congruent to 1 mod 6?
number-theory
asked Nov 19 at 6:58
paolo galli
223
add a comment |
up vote
0
down vote
favorite
A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.
In this question ec-numbers are introduced, obttained by the concatenation of two consecutive Mersenne numbers (12763=ec(7) for example). The values of $k$ for which $ec(k)$ is prime are: $[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]$
$7, 67, 360787$ are all the $k's$ leading to a prime ending with digit $7$. $7,67,360787$ are all congruent to $1$ $pmod 6$. My question is: does a k exist such that k ends with digit $7$, $ec(k)$ is prime and $k$ is NOT congruent to 1 $pmod 6$? Or that can be ruled out? In other words, if $k$ is congruent to 7 mod 10 and $ec(k)$ is prime, then necessarly $k$ must be congruent to 1 mod 6?
number-theory
asked Nov 19 at 6:58
paolo galli
223
I noticed that the largest factor of $6,66,360786$ is a palindrome. The largest factor of 6 is 3, the largest factor of 66 is 11, the largest factor of 360786 is 383.
– paolo galli
Nov 19 at 7:51
@Giovanni Resta can you find a counter-example?
– paolo galli
Nov 19 at 7:52
@Peter i think it is hard to find a counterexample, maybe it can be ruled out with a proof.
– paolo galli
Nov 19 at 8:15
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.
In this question ec-numbers are introduced, obttained by the concatenation of two consecutive Mersenne numbers (12763=ec(7) for example). The values of $k$ for which $ec(k)$ is prime are: $[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]$
$7, 67, 360787$ are all the $k's$ leading to a prime ending with digit $7$. $7,67,360787$ are all congruent to $1$ $pmod 6$. My question is: does a k exist such that k ends with digit $7$, $ec(k)$ is prime and $k$ is NOT congruent to 1 $pmod 6$? Or that can be ruled out? In other words, if $k$ is congruent to 7 mod 10 and $ec(k)$ is prime, then necessarly $k$ must be congruent to 1 mod 6?
number-theory
asked Nov 19 at 6:58
paolo galli
223
A conjecture about numbers of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.
In this question ec-numbers are introduced, obttained by the concatenation of two consecutive Mersenne numbers (12763=ec(7) for example). The values of $k$ for which $ec(k)$ is prime are: $[2, 3, 4, 7, 8, 12, 19, 22, 36, 46, 51, 67, 79, 215, 359, 394, 451, 1323, 2131, 3336, 3371, 6231, 19179, 39699, 51456, 56238, 69660, 75894, 79798, 92020, 174968, 176006, 181015, 285019, 331259, 360787, 366770]$
$7, 67, 360787$ are all the $k's$ leading to a prime ending with digit $7$. $7,67,360787$ are all congruent to $1$ $pmod 6$. My question is: does a k exist such that k ends with digit $7$, $ec(k)$ is prime and $k$ is NOT congruent to 1 $pmod 6$? Or that can be ruled out? In other words, if $k$ is congruent to 7 mod 10 and $ec(k)$ is prime, then necessarly $k$ must be congruent to 1 mod 6?
number-theory
number-theory
asked Nov 19 at 6:58
paolo galli
223
asked Nov 19 at 6:58
paolo galli
223
edited Nov 19 at 7:31
asked Nov 19 at 6:58
paolo galli
223
asked Nov 19 at 6:58
paolo galli
223
asked Nov 19 at 6:58
paolo galli
223
223
I noticed that the largest factor of $6,66,360786$ is a palindrome. The largest factor of 6 is 3, the largest factor of 66 is 11, the largest factor of 360786 is 383.
– paolo galli
Nov 19 at 7:51
@Giovanni Resta can you find a counter-example?
– paolo galli
Nov 19 at 7:52
@Peter i think it is hard to find a counterexample, maybe it can be ruled out with a proof.
– paolo galli
Nov 19 at 8:15
add a comment |
I noticed that the largest factor of $6,66,360786$ is a palindrome. The largest factor of 6 is 3, the largest factor of 66 is 11, the largest factor of 360786 is 383.
– paolo galli
Nov 19 at 7:51
@Giovanni Resta can you find a counter-example?
– paolo galli
Nov 19 at 7:52
@Peter i think it is hard to find a counterexample, maybe it can be ruled out with a proof.
– paolo galli
Nov 19 at 8:15
I noticed that the largest factor of $6,66,360786$ is a palindrome. The largest factor of 6 is 3, the largest factor of 66 is 11, the largest factor of 360786 is 383.
– paolo galli
Nov 19 at 7:51
I noticed that the largest factor of $6,66,360786$ is a palindrome. The largest factor of 6 is 3, the largest factor of 66 is 11, the largest factor of 360786 is 383.
– paolo galli
Nov 19 at 7:51
@Giovanni Resta can you find a counter-example?
– paolo galli
Nov 19 at 7:52
@Giovanni Resta can you find a counter-example?
– paolo galli
Nov 19 at 7:52
@Peter i think it is hard to find a counterexample, maybe it can be ruled out with a proof.
– paolo galli
Nov 19 at 8:15
@Peter i think it is hard to find a counterexample, maybe it can be ruled out with a proof.
– paolo galli
Nov 19 at 8:15
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004600%2fexponents-of-ec-primes-ending-with-digit-7%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
I noticed that the largest factor of $6,66,360786$ is a palindrome. The largest factor of 6 is 3, the largest factor of 66 is 11, the largest factor of 360786 is 383.
– paolo galli
Nov 19 at 7:51
@Giovanni Resta can you find a counter-example?
– paolo galli
Nov 19 at 7:52
@Peter i think it is hard to find a counterexample, maybe it can be ruled out with a proof.
– paolo galli
Nov 19 at 8:15