Find conditions on $a, b, c$, and $d$ with $ane -1, 0, 1$ such that $dmid(a^n+bn+c)$ for $n ge 1$.
This is a generalization of
Using induction, show that $4^n +15n - 1$ is divisible by $9$ for all $n geq 1$
I want to find conditions on
$a, b, c$, and $d$
with
$ane -1, 0, 1$
such that
$dmid(a^n+bn+c)$
for
$n ge 1$.
Here is my result:
A sufficient condition
is that
$a+b+c ne 0$
and
all of
$a+b+c,
b(a-1)$,
and
$c(a-1)-b$
are divisible by $d$.
For the problem
that prompted this,
with
$a=4, b=15, c=-1$,
these are
$18, 45,$
and
$-18$.
sequences-and-series elementary-number-theory divisibility
edited Nov 22 '18 at 10:47
Saad
19.7k92252
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
add a comment |
This is a generalization of
Using induction, show that $4^n +15n - 1$ is divisible by $9$ for all $n geq 1$
I want to find conditions on
$a, b, c$, and $d$
with
$ane -1, 0, 1$
such that
$dmid(a^n+bn+c)$
for
$n ge 1$.
Here is my result:
A sufficient condition
is that
$a+b+c ne 0$
and
all of
$a+b+c,
b(a-1)$,
and
$c(a-1)-b$
are divisible by $d$.
For the problem
that prompted this,
with
$a=4, b=15, c=-1$,
these are
$18, 45,$
and
$-18$.
sequences-and-series elementary-number-theory divisibility
edited Nov 22 '18 at 10:47
Saad
19.7k92252
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
1
This is a dupe (of at least a couple threads)
– Bill Dubuque
Nov 16 '18 at 4:29
Wouldn't be surprised. Might even be a dupe of myself, the way my memory works. Anyway, I worked this out just today completely independently. If you find the dupe, I'll upvote you. What the heck, I'll upvote you anyway.
– marty cohen
Nov 16 '18 at 5:11
2
I found a couple, e.g. here and here. There are likely more.
– Bill Dubuque
Nov 16 '18 at 15:35
I have done this as an excercise of induction. I think it will be hard to find the conditions.
– OppoInfinity
Nov 19 '18 at 4:33
add a comment |
This is a generalization of
Using induction, show that $4^n +15n - 1$ is divisible by $9$ for all $n geq 1$
I want to find conditions on
$a, b, c$, and $d$
with
$ane -1, 0, 1$
such that
$dmid(a^n+bn+c)$
for
$n ge 1$.
Here is my result:
A sufficient condition
is that
$a+b+c ne 0$
and
all of
$a+b+c,
b(a-1)$,
and
$c(a-1)-b$
are divisible by $d$.
For the problem
that prompted this,
with
$a=4, b=15, c=-1$,
these are
$18, 45,$
and
$-18$.
sequences-and-series elementary-number-theory divisibility
edited Nov 22 '18 at 10:47
Saad
19.7k92252
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
This is a generalization of
Using induction, show that $4^n +15n - 1$ is divisible by $9$ for all $n geq 1$
I want to find conditions on
$a, b, c$, and $d$
with
$ane -1, 0, 1$
such that
$dmid(a^n+bn+c)$
for
$n ge 1$.
Here is my result:
A sufficient condition
is that
$a+b+c ne 0$
and
all of
$a+b+c,
b(a-1)$,
and
$c(a-1)-b$
are divisible by $d$.
For the problem
that prompted this,
with
$a=4, b=15, c=-1$,
these are
$18, 45,$
and
$-18$.
sequences-and-series elementary-number-theory divisibility
sequences-and-series elementary-number-theory divisibility
edited Nov 22 '18 at 10:47
Saad
19.7k92252
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
edited Nov 22 '18 at 10:47
Saad
19.7k92252
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
edited Nov 22 '18 at 10:47
Saad
19.7k92252
edited Nov 22 '18 at 10:47
Saad
19.7k92252
edited Nov 22 '18 at 10:47
Saad
19.7k92252
19.7k92252
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
asked Nov 16 '18 at 3:56
marty cohen
72.5k549127
72.5k549127
1
This is a dupe (of at least a couple threads)
– Bill Dubuque
Nov 16 '18 at 4:29
Wouldn't be surprised. Might even be a dupe of myself, the way my memory works. Anyway, I worked this out just today completely independently. If you find the dupe, I'll upvote you. What the heck, I'll upvote you anyway.
– marty cohen
Nov 16 '18 at 5:11
2
I found a couple, e.g. here and here. There are likely more.
– Bill Dubuque
Nov 16 '18 at 15:35
I have done this as an excercise of induction. I think it will be hard to find the conditions.
– OppoInfinity
Nov 19 '18 at 4:33
add a comment |
1
This is a dupe (of at least a couple threads)
– Bill Dubuque
Nov 16 '18 at 4:29
Wouldn't be surprised. Might even be a dupe of myself, the way my memory works. Anyway, I worked this out just today completely independently. If you find the dupe, I'll upvote you. What the heck, I'll upvote you anyway.
– marty cohen
Nov 16 '18 at 5:11
2
I found a couple, e.g. here and here. There are likely more.
– Bill Dubuque
Nov 16 '18 at 15:35
I have done this as an excercise of induction. I think it will be hard to find the conditions.
– OppoInfinity
Nov 19 '18 at 4:33
1
1
This is a dupe (of at least a couple threads)
– Bill Dubuque
Nov 16 '18 at 4:29
This is a dupe (of at least a couple threads)
– Bill Dubuque
Nov 16 '18 at 4:29
Wouldn't be surprised. Might even be a dupe of myself, the way my memory works. Anyway, I worked this out just today completely independently. If you find the dupe, I'll upvote you. What the heck, I'll upvote you anyway.
– marty cohen
Nov 16 '18 at 5:11
Wouldn't be surprised. Might even be a dupe of myself, the way my memory works. Anyway, I worked this out just today completely independently. If you find the dupe, I'll upvote you. What the heck, I'll upvote you anyway.
– marty cohen
Nov 16 '18 at 5:11
2
2
I found a couple, e.g. here and here. There are likely more.
– Bill Dubuque
Nov 16 '18 at 15:35
I found a couple, e.g. here and here. There are likely more.
– Bill Dubuque
Nov 16 '18 at 15:35
I have done this as an excercise of induction. I think it will be hard to find the conditions.
– OppoInfinity
Nov 19 '18 at 4:33
I have done this as an excercise of induction. I think it will be hard to find the conditions.
– OppoInfinity
Nov 19 '18 at 4:33
add a comment |
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1
This is a dupe (of at least a couple threads)
– Bill Dubuque
Nov 16 '18 at 4:29
Wouldn't be surprised. Might even be a dupe of myself, the way my memory works. Anyway, I worked this out just today completely independently. If you find the dupe, I'll upvote you. What the heck, I'll upvote you anyway.
– marty cohen
Nov 16 '18 at 5:11
2
I found a couple, e.g. here and here. There are likely more.
– Bill Dubuque
Nov 16 '18 at 15:35
I have done this as an excercise of induction. I think it will be hard to find the conditions.
– OppoInfinity
Nov 19 '18 at 4:33