Prove that every prime number divides some number in the sequence
$begingroup$
Let ($a_n$) be a sequence of numbers such that for all natural numbers n:
$a_n = 2^n+3^n+6^n-1$
Show that every prime number divides some number in that sequence.
linear-algebra sequences-and-series prime-numbers recurrence-relations
$endgroup$
add a comment |
$begingroup$
Let ($a_n$) be a sequence of numbers such that for all natural numbers n:
$a_n = 2^n+3^n+6^n-1$
Show that every prime number divides some number in that sequence.
linear-algebra sequences-and-series prime-numbers recurrence-relations
$endgroup$
$begingroup$
I trust you have made some effort to solve this on your own. Please show this in your question, including what you've had difficulty with. Thanks.
$endgroup$
– John Omielan
40 mins ago
add a comment |
$begingroup$
Let ($a_n$) be a sequence of numbers such that for all natural numbers n:
$a_n = 2^n+3^n+6^n-1$
Show that every prime number divides some number in that sequence.
linear-algebra sequences-and-series prime-numbers recurrence-relations
$endgroup$
Let ($a_n$) be a sequence of numbers such that for all natural numbers n:
$a_n = 2^n+3^n+6^n-1$
Show that every prime number divides some number in that sequence.
linear-algebra sequences-and-series prime-numbers recurrence-relations
linear-algebra sequences-and-series prime-numbers recurrence-relations
asked 48 mins ago
DoodDood
357
357
$begingroup$
I trust you have made some effort to solve this on your own. Please show this in your question, including what you've had difficulty with. Thanks.
$endgroup$
– John Omielan
40 mins ago
add a comment |
$begingroup$
I trust you have made some effort to solve this on your own. Please show this in your question, including what you've had difficulty with. Thanks.
$endgroup$
– John Omielan
40 mins ago
$begingroup$
I trust you have made some effort to solve this on your own. Please show this in your question, including what you've had difficulty with. Thanks.
$endgroup$
– John Omielan
40 mins ago
$begingroup$
I trust you have made some effort to solve this on your own. Please show this in your question, including what you've had difficulty with. Thanks.
$endgroup$
– John Omielan
40 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $p$ be prime number. Consider $a_{p-2}=2^{p-2}+3^{p-2}+6^{p-2}-1$. Then
begin{align*}
6a_{p-2} & equiv 6(2^{p-2}+3^{p-2}+6^{p-2})-6 pmod{p}\
& equiv 3cdot 2^{p-1}+2cdot 3^{p-1}+6^{p-1}-6pmod{p}\
& equiv 3+2+1 -6pmod{p}\
& equiv 0 pmod{p}.
end{align*}
For $p>3$, we will have $gcd(6,p)=1$. So we can multiply by $6^{-1}$ on both sides to get
$$a_{p-2} equiv 0 pmod{p}$$
Now you can deal with $p=2,3$ as special case.
$endgroup$
3
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
1
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
add a comment |
Your Answer
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1 Answer
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active
oldest
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1 Answer
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active
oldest
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active
oldest
votes
$begingroup$
Let $p$ be prime number. Consider $a_{p-2}=2^{p-2}+3^{p-2}+6^{p-2}-1$. Then
begin{align*}
6a_{p-2} & equiv 6(2^{p-2}+3^{p-2}+6^{p-2})-6 pmod{p}\
& equiv 3cdot 2^{p-1}+2cdot 3^{p-1}+6^{p-1}-6pmod{p}\
& equiv 3+2+1 -6pmod{p}\
& equiv 0 pmod{p}.
end{align*}
For $p>3$, we will have $gcd(6,p)=1$. So we can multiply by $6^{-1}$ on both sides to get
$$a_{p-2} equiv 0 pmod{p}$$
Now you can deal with $p=2,3$ as special case.
$endgroup$
3
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
1
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
add a comment |
$begingroup$
Let $p$ be prime number. Consider $a_{p-2}=2^{p-2}+3^{p-2}+6^{p-2}-1$. Then
begin{align*}
6a_{p-2} & equiv 6(2^{p-2}+3^{p-2}+6^{p-2})-6 pmod{p}\
& equiv 3cdot 2^{p-1}+2cdot 3^{p-1}+6^{p-1}-6pmod{p}\
& equiv 3+2+1 -6pmod{p}\
& equiv 0 pmod{p}.
end{align*}
For $p>3$, we will have $gcd(6,p)=1$. So we can multiply by $6^{-1}$ on both sides to get
$$a_{p-2} equiv 0 pmod{p}$$
Now you can deal with $p=2,3$ as special case.
$endgroup$
3
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
1
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
add a comment |
$begingroup$
Let $p$ be prime number. Consider $a_{p-2}=2^{p-2}+3^{p-2}+6^{p-2}-1$. Then
begin{align*}
6a_{p-2} & equiv 6(2^{p-2}+3^{p-2}+6^{p-2})-6 pmod{p}\
& equiv 3cdot 2^{p-1}+2cdot 3^{p-1}+6^{p-1}-6pmod{p}\
& equiv 3+2+1 -6pmod{p}\
& equiv 0 pmod{p}.
end{align*}
For $p>3$, we will have $gcd(6,p)=1$. So we can multiply by $6^{-1}$ on both sides to get
$$a_{p-2} equiv 0 pmod{p}$$
Now you can deal with $p=2,3$ as special case.
$endgroup$
Let $p$ be prime number. Consider $a_{p-2}=2^{p-2}+3^{p-2}+6^{p-2}-1$. Then
begin{align*}
6a_{p-2} & equiv 6(2^{p-2}+3^{p-2}+6^{p-2})-6 pmod{p}\
& equiv 3cdot 2^{p-1}+2cdot 3^{p-1}+6^{p-1}-6pmod{p}\
& equiv 3+2+1 -6pmod{p}\
& equiv 0 pmod{p}.
end{align*}
For $p>3$, we will have $gcd(6,p)=1$. So we can multiply by $6^{-1}$ on both sides to get
$$a_{p-2} equiv 0 pmod{p}$$
Now you can deal with $p=2,3$ as special case.
edited 32 mins ago
Robert Israel
321k23210462
321k23210462
answered 33 mins ago
Anurag AAnurag A
25.9k12249
25.9k12249
3
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
1
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
add a comment |
3
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
1
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
3
3
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
$begingroup$
Intuitively, taking $n=-1$ would give $a_{-1} = frac12 + frac13 + frac16 - 1 = 0$, and the remainders mod $p$ should repeat every $p-1$ steps, so taking $n=p-2$ also works. This isn't quite rigorous, but it's motivation for considering $a_{p-2}$.
$endgroup$
– Misha Lavrov
31 mins ago
1
1
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
$begingroup$
Thanks, I had to look up how to justify that all these powers were congruent to 1 (mod.p). Indeed Fermats little theorem solves this problem fast. Thanks for help and for such quick response.
$endgroup$
– Dood
27 mins ago
add a comment |
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$begingroup$
I trust you have made some effort to solve this on your own. Please show this in your question, including what you've had difficulty with. Thanks.
$endgroup$
– John Omielan
40 mins ago