Fermat's Little Theorem and Carmichael Numbers
Fermat's little theorem states that if $p$ is a prime number and $a$ is a positive integer, then $p|a^p-a$.
However, the converse is false, that is, for integers $a$ and $p$, if $p|a^p-a$, then $a$ is a prime number, is a false statement. For instant, $561|a^{561}-a$ for some integer $a$, but $561$ is actually a composite number, and such numbers are called "Carmichael numbers".
In other words, a Carmichael number is a composite integer, say $k$, such that $k|a^k-a$ for all integers $a$.
This is what I know, am I right or I misunderstand something?
and
Do we have a way to find Carmichael numbers?
number-theory elementary-number-theory prime-numbers carmichael-numbers
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
add a comment |
Fermat's little theorem states that if $p$ is a prime number and $a$ is a positive integer, then $p|a^p-a$.
However, the converse is false, that is, for integers $a$ and $p$, if $p|a^p-a$, then $a$ is a prime number, is a false statement. For instant, $561|a^{561}-a$ for some integer $a$, but $561$ is actually a composite number, and such numbers are called "Carmichael numbers".
In other words, a Carmichael number is a composite integer, say $k$, such that $k|a^k-a$ for all integers $a$.
This is what I know, am I right or I misunderstand something?
and
Do we have a way to find Carmichael numbers?
number-theory elementary-number-theory prime-numbers carmichael-numbers
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
2
Search the web for find Carmicael numbers maths.lancs.ac.uk/jameson/carfind.pdf
– Ethan Bolker
Nov 28 '18 at 17:04
There is a nice criterion for a squarefree odd composite $N>1$ : $N$ is a Carmichael number if and only if $p-1mid N-1$ holds for every prime $pmid N$. If $N$ is not squarefree odd and composite, it cannot be a Carmichael number. Moreover, it can be shown that $N$ must have at least three prime factors.
– Peter
Nov 28 '18 at 19:00
add a comment |
Fermat's little theorem states that if $p$ is a prime number and $a$ is a positive integer, then $p|a^p-a$.
However, the converse is false, that is, for integers $a$ and $p$, if $p|a^p-a$, then $a$ is a prime number, is a false statement. For instant, $561|a^{561}-a$ for some integer $a$, but $561$ is actually a composite number, and such numbers are called "Carmichael numbers".
In other words, a Carmichael number is a composite integer, say $k$, such that $k|a^k-a$ for all integers $a$.
This is what I know, am I right or I misunderstand something?
and
Do we have a way to find Carmichael numbers?
number-theory elementary-number-theory prime-numbers carmichael-numbers
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
Fermat's little theorem states that if $p$ is a prime number and $a$ is a positive integer, then $p|a^p-a$.
However, the converse is false, that is, for integers $a$ and $p$, if $p|a^p-a$, then $a$ is a prime number, is a false statement. For instant, $561|a^{561}-a$ for some integer $a$, but $561$ is actually a composite number, and such numbers are called "Carmichael numbers".
In other words, a Carmichael number is a composite integer, say $k$, such that $k|a^k-a$ for all integers $a$.
This is what I know, am I right or I misunderstand something?
and
Do we have a way to find Carmichael numbers?
number-theory elementary-number-theory prime-numbers carmichael-numbers
number-theory elementary-number-theory prime-numbers carmichael-numbers
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
asked Nov 28 '18 at 17:02
Hussain-Alqatari
2997
2997
2
Search the web for find Carmicael numbers maths.lancs.ac.uk/jameson/carfind.pdf
– Ethan Bolker
Nov 28 '18 at 17:04
There is a nice criterion for a squarefree odd composite $N>1$ : $N$ is a Carmichael number if and only if $p-1mid N-1$ holds for every prime $pmid N$. If $N$ is not squarefree odd and composite, it cannot be a Carmichael number. Moreover, it can be shown that $N$ must have at least three prime factors.
– Peter
Nov 28 '18 at 19:00
add a comment |
2
Search the web for find Carmicael numbers maths.lancs.ac.uk/jameson/carfind.pdf
– Ethan Bolker
Nov 28 '18 at 17:04
There is a nice criterion for a squarefree odd composite $N>1$ : $N$ is a Carmichael number if and only if $p-1mid N-1$ holds for every prime $pmid N$. If $N$ is not squarefree odd and composite, it cannot be a Carmichael number. Moreover, it can be shown that $N$ must have at least three prime factors.
– Peter
Nov 28 '18 at 19:00
2
2
Search the web for find Carmicael numbers maths.lancs.ac.uk/jameson/carfind.pdf
– Ethan Bolker
Nov 28 '18 at 17:04
Search the web for find Carmicael numbers maths.lancs.ac.uk/jameson/carfind.pdf
– Ethan Bolker
Nov 28 '18 at 17:04
There is a nice criterion for a squarefree odd composite $N>1$ : $N$ is a Carmichael number if and only if $p-1mid N-1$ holds for every prime $pmid N$. If $N$ is not squarefree odd and composite, it cannot be a Carmichael number. Moreover, it can be shown that $N$ must have at least three prime factors.
– Peter
Nov 28 '18 at 19:00
There is a nice criterion for a squarefree odd composite $N>1$ : $N$ is a Carmichael number if and only if $p-1mid N-1$ holds for every prime $pmid N$. If $N$ is not squarefree odd and composite, it cannot be a Carmichael number. Moreover, it can be shown that $N$ must have at least three prime factors.
– Peter
Nov 28 '18 at 19:00
add a comment |
1 Answer
1
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oldest
votes
Numbers of the form $(6k+1)(12k+1)(18k+1)$ are Carmichael numbers if each of the three factors is prime. This gives some examples - already $k=1$ works. Actually, the sequence A046025 gives more values, e.g.,
$$
k=1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380,
426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115,
1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421,
1441, 1490, 1515, 1696, 1805, 1875, 1885
$$
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
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oldest
votes
active
oldest
votes
Numbers of the form $(6k+1)(12k+1)(18k+1)$ are Carmichael numbers if each of the three factors is prime. This gives some examples - already $k=1$ works. Actually, the sequence A046025 gives more values, e.g.,
$$
k=1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380,
426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115,
1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421,
1441, 1490, 1515, 1696, 1805, 1875, 1885
$$
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
add a comment |
Numbers of the form $(6k+1)(12k+1)(18k+1)$ are Carmichael numbers if each of the three factors is prime. This gives some examples - already $k=1$ works. Actually, the sequence A046025 gives more values, e.g.,
$$
k=1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380,
426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115,
1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421,
1441, 1490, 1515, 1696, 1805, 1875, 1885
$$
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
add a comment |
Numbers of the form $(6k+1)(12k+1)(18k+1)$ are Carmichael numbers if each of the three factors is prime. This gives some examples - already $k=1$ works. Actually, the sequence A046025 gives more values, e.g.,
$$
k=1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380,
426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115,
1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421,
1441, 1490, 1515, 1696, 1805, 1875, 1885
$$
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
Numbers of the form $(6k+1)(12k+1)(18k+1)$ are Carmichael numbers if each of the three factors is prime. This gives some examples - already $k=1$ works. Actually, the sequence A046025 gives more values, e.g.,
$$
k=1, 6, 35, 45, 51, 55, 56, 100, 121, 195, 206, 216, 255, 276, 370, 380,
426, 506, 510, 511, 710, 741, 800, 825, 871, 930, 975, 1025, 1060, 1115,
1140, 1161, 1270, 1280, 1281, 1311, 1336, 1361, 1365, 1381, 1420, 1421,
1441, 1490, 1515, 1696, 1805, 1875, 1885
$$
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
answered Nov 28 '18 at 19:13
Dietrich Burde
77.9k64386
77.9k64386
add a comment |
add a comment |
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2
Search the web for find Carmicael numbers maths.lancs.ac.uk/jameson/carfind.pdf
– Ethan Bolker
Nov 28 '18 at 17:04
There is a nice criterion for a squarefree odd composite $N>1$ : $N$ is a Carmichael number if and only if $p-1mid N-1$ holds for every prime $pmid N$. If $N$ is not squarefree odd and composite, it cannot be a Carmichael number. Moreover, it can be shown that $N$ must have at least three prime factors.
– Peter
Nov 28 '18 at 19:00