Product of arithmetic progressions
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Let $(a_1,a_2ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,dots,a_nb_n)$ is an arithmetic progression?
The sequence is (trivially) an arithmetic progression when $n=1$ or $2$. We can notice that $(a_1,a_2,ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ cannot be in identical order, since that would mean they both have to be increasing or decreasing, but then the differences of the resulting product terms cannot be constant.
arithmetic-progressions
asked Sep 9 at 12:17
pi66
3,9291037
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up vote
3
down vote
favorite
Let $(a_1,a_2ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,dots,a_nb_n)$ is an arithmetic progression?
The sequence is (trivially) an arithmetic progression when $n=1$ or $2$. We can notice that $(a_1,a_2,ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ cannot be in identical order, since that would mean they both have to be increasing or decreasing, but then the differences of the resulting product terms cannot be constant.
arithmetic-progressions
asked Sep 9 at 12:17
pi66
3,9291037
1
11,5,8 and 2,3,1 work for n=3
– Empy2
Sep 9 at 13:27
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $(a_1,a_2ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,dots,a_nb_n)$ is an arithmetic progression?
The sequence is (trivially) an arithmetic progression when $n=1$ or $2$. We can notice that $(a_1,a_2,ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ cannot be in identical order, since that would mean they both have to be increasing or decreasing, but then the differences of the resulting product terms cannot be constant.
arithmetic-progressions
asked Sep 9 at 12:17
pi66
3,9291037
Let $(a_1,a_2ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,dots,a_nb_n)$ is an arithmetic progression?
The sequence is (trivially) an arithmetic progression when $n=1$ or $2$. We can notice that $(a_1,a_2,ldots,a_n)$ and $(b_1,b_2,ldots,b_n)$ cannot be in identical order, since that would mean they both have to be increasing or decreasing, but then the differences of the resulting product terms cannot be constant.
arithmetic-progressions
arithmetic-progressions
asked Sep 9 at 12:17
pi66
3,9291037
asked Sep 9 at 12:17
pi66
3,9291037
asked Sep 9 at 12:17
pi66
3,9291037
asked Sep 9 at 12:17
pi66
3,9291037
asked Sep 9 at 12:17
pi66
3,9291037
3,9291037
1
11,5,8 and 2,3,1 work for n=3
– Empy2
Sep 9 at 13:27
add a comment |Â
1
11,5,8 and 2,3,1 work for n=3
– Empy2
Sep 9 at 13:27
1
1
11,5,8 and 2,3,1 work for n=3
– Empy2
Sep 9 at 13:27
11,5,8 and 2,3,1 work for n=3
– Empy2
Sep 9 at 13:27
add a comment |Â
1 Answer
1
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My computer found these for 4, 5 and 6:
$$A: 1×10, 11×4, 6×13, 16×7\
B: 8×4, 6×9, 4×19, 7×14, 5×24\
C: 7×35, 31×11,19×23,13×41,37×17,25×29$$
answered Sep 10 at 0:35
Empy2
32.2k12059
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
My computer found these for 4, 5 and 6:
$$A: 1×10, 11×4, 6×13, 16×7\
B: 8×4, 6×9, 4×19, 7×14, 5×24\
C: 7×35, 31×11,19×23,13×41,37×17,25×29$$
answered Sep 10 at 0:35
Empy2
32.2k12059
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
add a comment |Â
up vote
0
down vote
My computer found these for 4, 5 and 6:
$$A: 1×10, 11×4, 6×13, 16×7\
B: 8×4, 6×9, 4×19, 7×14, 5×24\
C: 7×35, 31×11,19×23,13×41,37×17,25×29$$
answered Sep 10 at 0:35
Empy2
32.2k12059
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
add a comment |Â
up vote
0
down vote
up vote
0
down vote
My computer found these for 4, 5 and 6:
$$A: 1×10, 11×4, 6×13, 16×7\
B: 8×4, 6×9, 4×19, 7×14, 5×24\
C: 7×35, 31×11,19×23,13×41,37×17,25×29$$
answered Sep 10 at 0:35
Empy2
32.2k12059
My computer found these for 4, 5 and 6:
$$A: 1×10, 11×4, 6×13, 16×7\
B: 8×4, 6×9, 4×19, 7×14, 5×24\
C: 7×35, 31×11,19×23,13×41,37×17,25×29$$
answered Sep 10 at 0:35
Empy2
32.2k12059
edited Sep 13 at 5:38
answered Sep 10 at 0:35
Empy2
32.2k12059
answered Sep 10 at 0:35
Empy2
32.2k12059
answered Sep 10 at 0:35
Empy2
32.2k12059
32.2k12059
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
add a comment |Â
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
$4×41,6×31,8×26,5×46,7×36$
– Empy2
Sep 12 at 10:25
add a comment |Â
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1
11,5,8 and 2,3,1 work for n=3
– Empy2
Sep 9 at 13:27