Why 142857 is special??For which number does multiplying it by 99 add a 1 to each end of its decimal representation?Find 7 digit prime numbers with this property;A problem for math lovers to count the digitsDoes there exist $n$ such that all numbers $n,2n,dots,2000n$ have the same digits?Sudoku with special propertiesIs there any other special numbers?About $142857$: proof that $;3 mid 1^n + 4^n + 2^n + 8^n + 5^n + 7^n$Special properties of the number $146$
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Why 142857 is special??
For which number does multiplying it by 99 add a 1 to each end of its decimal representation?Find 7 digit prime numbers with this property;A problem for math lovers to count the digitsDoes there exist $n$ such that all numbers $n,2n,dots,2000n$ have the same digits?Sudoku with special propertiesIs there any other special numbers?About $142857$: proof that $;3 mid 1^n + 4^n + 2^n + 8^n + 5^n + 7^n$Special properties of the number $146$
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;
$begingroup$
What made 142857 a special number?
Why it gives the same digits if it is multiplied by 1,2,3,4,5 & 6 ?
And gives all nines when it is multiplied by 7?
recreational-mathematics mathematica
edited 8 hours ago
Derek Elkins
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asked 9 hours ago
user698179user698179
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
add a comment |
$begingroup$
What made 142857 a special number?
Why it gives the same digits if it is multiplied by 1,2,3,4,5 & 6 ?
And gives all nines when it is multiplied by 7?
recreational-mathematics mathematica
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
asked 9 hours ago
user698179user698179
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Please include source/motivation for this question.
$endgroup$
– tatan
9 hours ago
$begingroup$
It just got struck in my head... And it started as a puzzle to me
$endgroup$
– user698179
9 hours ago
$begingroup$
It's $(10^6-1)/7$.
$endgroup$
– Lord Shark the Unknown
8 hours ago
add a comment |
$begingroup$
What made 142857 a special number?
Why it gives the same digits if it is multiplied by 1,2,3,4,5 & 6 ?
And gives all nines when it is multiplied by 7?
recreational-mathematics mathematica
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
asked 9 hours ago
user698179user698179
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
What made 142857 a special number?
Why it gives the same digits if it is multiplied by 1,2,3,4,5 & 6 ?
And gives all nines when it is multiplied by 7?
recreational-mathematics mathematica
recreational-mathematics mathematica
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
asked 9 hours ago
user698179user698179
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
asked 9 hours ago
user698179user698179
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
edited 8 hours ago
Derek Elkins
20.4k1 gold badge16 silver badges39 bronze badges
20.4k1 gold badge16 silver badges39 bronze badges
asked 9 hours ago
user698179user698179
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 9 hours ago
user698179user698179
233 bronze badges
asked 9 hours ago
user698179user698179
233 bronze badges
233 bronze badges
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
user698179 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Please include source/motivation for this question.
$endgroup$
– tatan
9 hours ago
$begingroup$
It just got struck in my head... And it started as a puzzle to me
$endgroup$
– user698179
9 hours ago
$begingroup$
It's $(10^6-1)/7$.
$endgroup$
– Lord Shark the Unknown
8 hours ago
add a comment |
$begingroup$
Please include source/motivation for this question.
$endgroup$
– tatan
9 hours ago
$begingroup$
It just got struck in my head... And it started as a puzzle to me
$endgroup$
– user698179
9 hours ago
$begingroup$
It's $(10^6-1)/7$.
$endgroup$
– Lord Shark the Unknown
8 hours ago
$begingroup$
Please include source/motivation for this question.
$endgroup$
– tatan
9 hours ago
$begingroup$
Please include source/motivation for this question.
$endgroup$
– tatan
9 hours ago
$begingroup$
It just got struck in my head... And it started as a puzzle to me
$endgroup$
– user698179
9 hours ago
$begingroup$
It just got struck in my head... And it started as a puzzle to me
$endgroup$
– user698179
9 hours ago
$begingroup$
It's $(10^6-1)/7$.
$endgroup$
– Lord Shark the Unknown
8 hours ago
$begingroup$
It's $(10^6-1)/7$.
$endgroup$
– Lord Shark the Unknown
8 hours ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
More generally, this happens for the fraction $1/n$ exactly when $10$ is a primitive root mod $n$.
Those $n$ are the ones in A167797:
$$
7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, dots
$$
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
$endgroup$
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
add a comment |
$begingroup$
Because
$dfrac17
=.142857142857...
$
and all
(and there is a lot)
that follows from that.
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
$endgroup$
add a comment |
$begingroup$
A very interesting fact about $142857$ besides $1over 7=0.overline142857$ is that the digits of $ntimes 142857$ are always a permutation of $142857$ itself for $1le nle 6=7-1$.
$$1times 142857=142857\2times 142857=285714\3times 142857=428571\4times 142857=571428\5times 142857=714285\6times 142857=857142$$
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
$endgroup$
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
More generally, this happens for the fraction $1/n$ exactly when $10$ is a primitive root mod $n$.
Those $n$ are the ones in A167797:
$$
7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, dots
$$
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
$endgroup$
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
add a comment |
$begingroup$
More generally, this happens for the fraction $1/n$ exactly when $10$ is a primitive root mod $n$.
Those $n$ are the ones in A167797:
$$
7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, dots
$$
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
$endgroup$
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
add a comment |
$begingroup$
More generally, this happens for the fraction $1/n$ exactly when $10$ is a primitive root mod $n$.
Those $n$ are the ones in A167797:
$$
7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, dots
$$
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
$endgroup$
More generally, this happens for the fraction $1/n$ exactly when $10$ is a primitive root mod $n$.
Those $n$ are the ones in A167797:
$$
7, 17, 19, 23, 29, 47, 49, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, dots
$$
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
answered 8 hours ago
lhflhf
173k11 gold badges179 silver badges418 bronze badges
173k11 gold badges179 silver badges418 bronze badges
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
add a comment |
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Full_reptend_prime
$endgroup$
– lhf
8 hours ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
$begingroup$
See also en.wikipedia.org/wiki/Cyclic_number
$endgroup$
– lhf
1 hour ago
add a comment |
$begingroup$
Because
$dfrac17
=.142857142857...
$
and all
(and there is a lot)
that follows from that.
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
$endgroup$
add a comment |
$begingroup$
Because
$dfrac17
=.142857142857...
$
and all
(and there is a lot)
that follows from that.
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
$endgroup$
add a comment |
$begingroup$
Because
$dfrac17
=.142857142857...
$
and all
(and there is a lot)
that follows from that.
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
$endgroup$
Because
$dfrac17
=.142857142857...
$
and all
(and there is a lot)
that follows from that.
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
answered 9 hours ago
marty cohenmarty cohen
79.5k5 gold badges52 silver badges134 bronze badges
79.5k5 gold badges52 silver badges134 bronze badges
add a comment |
add a comment |
$begingroup$
A very interesting fact about $142857$ besides $1over 7=0.overline142857$ is that the digits of $ntimes 142857$ are always a permutation of $142857$ itself for $1le nle 6=7-1$.
$$1times 142857=142857\2times 142857=285714\3times 142857=428571\4times 142857=571428\5times 142857=714285\6times 142857=857142$$
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
$endgroup$
add a comment |
$begingroup$
A very interesting fact about $142857$ besides $1over 7=0.overline142857$ is that the digits of $ntimes 142857$ are always a permutation of $142857$ itself for $1le nle 6=7-1$.
$$1times 142857=142857\2times 142857=285714\3times 142857=428571\4times 142857=571428\5times 142857=714285\6times 142857=857142$$
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
$endgroup$
add a comment |
$begingroup$
A very interesting fact about $142857$ besides $1over 7=0.overline142857$ is that the digits of $ntimes 142857$ are always a permutation of $142857$ itself for $1le nle 6=7-1$.
$$1times 142857=142857\2times 142857=285714\3times 142857=428571\4times 142857=571428\5times 142857=714285\6times 142857=857142$$
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
$endgroup$
A very interesting fact about $142857$ besides $1over 7=0.overline142857$ is that the digits of $ntimes 142857$ are always a permutation of $142857$ itself for $1le nle 6=7-1$.
$$1times 142857=142857\2times 142857=285714\3times 142857=428571\4times 142857=571428\5times 142857=714285\6times 142857=857142$$
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
answered 7 hours ago
Mostafa AyazMostafa Ayaz
19.5k3 gold badges10 silver badges42 bronze badges
19.5k3 gold badges10 silver badges42 bronze badges
add a comment |
add a comment |
user698179 is a new contributor. Be nice, and check out our Code of Conduct.
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user698179 is a new contributor. Be nice, and check out our Code of Conduct.
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$begingroup$
Please include source/motivation for this question.
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– tatan
9 hours ago
$begingroup$
It just got struck in my head... And it started as a puzzle to me
$endgroup$
– user698179
9 hours ago
$begingroup$
It's $(10^6-1)/7$.
$endgroup$
– Lord Shark the Unknown
8 hours ago