Mixing
can we find multiple of a set of finite numbers that are in the “middle” of numbers mod a prime
$begingroup$
$newcommandN{mathbb N} newcommandceil[1]{lceil#1rceil}$Let $a_1,dots, a_kin N$ be an arbitrary finite set of positive integers. Can we find a prime number $p$ such that $p>k$ (preferably $pgg k$) and a natural number $ninN$ such that
$$a_1n,a_2n,dots,a_kn$$
are equivalent to integers in $[p/k,p-p/k]$ mod $p$?
Note: $[a,b]$ here just means a closed interval in the real line.
combinatorics number-theory elementary-number-theory discrete-mathematics
asked Jan 26 at 8:14
quantumquantum
538210
$endgroup$
add a comment |
$begingroup$
$newcommandN{mathbb N} newcommandceil[1]{lceil#1rceil}$Let $a_1,dots, a_kin N$ be an arbitrary finite set of positive integers. Can we find a prime number $p$ such that $p>k$ (preferably $pgg k$) and a natural number $ninN$ such that
$$a_1n,a_2n,dots,a_kn$$
are equivalent to integers in $[p/k,p-p/k]$ mod $p$?
Note: $[a,b]$ here just means a closed interval in the real line.
combinatorics number-theory elementary-number-theory discrete-mathematics
asked Jan 26 at 8:14
quantumquantum
538210
$endgroup$
$begingroup$
What does [a,b] mean?
$endgroup$
– William Elliot
Jan 26 at 9:45
$begingroup$
a closed interval in the real line
$endgroup$
– quantum
Jan 26 at 10:43
add a comment |
$begingroup$
$newcommandN{mathbb N} newcommandceil[1]{lceil#1rceil}$Let $a_1,dots, a_kin N$ be an arbitrary finite set of positive integers. Can we find a prime number $p$ such that $p>k$ (preferably $pgg k$) and a natural number $ninN$ such that
$$a_1n,a_2n,dots,a_kn$$
are equivalent to integers in $[p/k,p-p/k]$ mod $p$?
Note: $[a,b]$ here just means a closed interval in the real line.
combinatorics number-theory elementary-number-theory discrete-mathematics
asked Jan 26 at 8:14
quantumquantum
538210
$endgroup$
$newcommandN{mathbb N} newcommandceil[1]{lceil#1rceil}$Let $a_1,dots, a_kin N$ be an arbitrary finite set of positive integers. Can we find a prime number $p$ such that $p>k$ (preferably $pgg k$) and a natural number $ninN$ such that
$$a_1n,a_2n,dots,a_kn$$
are equivalent to integers in $[p/k,p-p/k]$ mod $p$?
Note: $[a,b]$ here just means a closed interval in the real line.
combinatorics number-theory elementary-number-theory discrete-mathematics
combinatorics number-theory elementary-number-theory discrete-mathematics
asked Jan 26 at 8:14
quantumquantum
538210
asked Jan 26 at 8:14
quantumquantum
538210
edited Jan 26 at 10:49
quantum
asked Jan 26 at 8:14
quantumquantum
538210
asked Jan 26 at 8:14
quantumquantum
538210
asked Jan 26 at 8:14
quantumquantum
538210
538210
$begingroup$
What does [a,b] mean?
$endgroup$
– William Elliot
Jan 26 at 9:45
$begingroup$
a closed interval in the real line
$endgroup$
– quantum
Jan 26 at 10:43
add a comment |
$begingroup$
What does [a,b] mean?
$endgroup$
– William Elliot
Jan 26 at 9:45
$begingroup$
a closed interval in the real line
$endgroup$
– quantum
Jan 26 at 10:43
$begingroup$
What does [a,b] mean?
$endgroup$
– William Elliot
Jan 26 at 9:45
$begingroup$
What does [a,b] mean?
$endgroup$
– William Elliot
Jan 26 at 9:45
$begingroup$
a closed interval in the real line
$endgroup$
– quantum
Jan 26 at 10:43
$begingroup$
a closed interval in the real line
$endgroup$
– quantum
Jan 26 at 10:43
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Not always. Indeed, let $a_i=i$ for each $i$ and $n$ be any natural number distinct from $1$ and $p$.
Since there are $p$ residues modulo $p$ and $k+1$ numbers of the form $ni$ for $i=0,dots, k$, by pigeonhole principle, there exist $0le i<jle k$ and an integer $Delta$ with $|Delta|lefrac p{k+1} $ such that $jnequiv in+Deltapmod p$. Then $0<j-ile k$ and $(j-i)n$ is equivalent mod $p$ to a unique integer in $[0,p-1]$ which is $|Delta|lefrac p{k+1}<frac p{k}$ or $p-|Delta|ge p-frac p{k+1}>p-frac p{k}$.
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
add a comment |
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1 Answer
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$begingroup$
Not always. Indeed, let $a_i=i$ for each $i$ and $n$ be any natural number distinct from $1$ and $p$.
Since there are $p$ residues modulo $p$ and $k+1$ numbers of the form $ni$ for $i=0,dots, k$, by pigeonhole principle, there exist $0le i<jle k$ and an integer $Delta$ with $|Delta|lefrac p{k+1} $ such that $jnequiv in+Deltapmod p$. Then $0<j-ile k$ and $(j-i)n$ is equivalent mod $p$ to a unique integer in $[0,p-1]$ which is $|Delta|lefrac p{k+1}<frac p{k}$ or $p-|Delta|ge p-frac p{k+1}>p-frac p{k}$.
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
add a comment |
$begingroup$
Not always. Indeed, let $a_i=i$ for each $i$ and $n$ be any natural number distinct from $1$ and $p$.
Since there are $p$ residues modulo $p$ and $k+1$ numbers of the form $ni$ for $i=0,dots, k$, by pigeonhole principle, there exist $0le i<jle k$ and an integer $Delta$ with $|Delta|lefrac p{k+1} $ such that $jnequiv in+Deltapmod p$. Then $0<j-ile k$ and $(j-i)n$ is equivalent mod $p$ to a unique integer in $[0,p-1]$ which is $|Delta|lefrac p{k+1}<frac p{k}$ or $p-|Delta|ge p-frac p{k+1}>p-frac p{k}$.
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
add a comment |
$begingroup$
Not always. Indeed, let $a_i=i$ for each $i$ and $n$ be any natural number distinct from $1$ and $p$.
Since there are $p$ residues modulo $p$ and $k+1$ numbers of the form $ni$ for $i=0,dots, k$, by pigeonhole principle, there exist $0le i<jle k$ and an integer $Delta$ with $|Delta|lefrac p{k+1} $ such that $jnequiv in+Deltapmod p$. Then $0<j-ile k$ and $(j-i)n$ is equivalent mod $p$ to a unique integer in $[0,p-1]$ which is $|Delta|lefrac p{k+1}<frac p{k}$ or $p-|Delta|ge p-frac p{k+1}>p-frac p{k}$.
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
$endgroup$
Not always. Indeed, let $a_i=i$ for each $i$ and $n$ be any natural number distinct from $1$ and $p$.
Since there are $p$ residues modulo $p$ and $k+1$ numbers of the form $ni$ for $i=0,dots, k$, by pigeonhole principle, there exist $0le i<jle k$ and an integer $Delta$ with $|Delta|lefrac p{k+1} $ such that $jnequiv in+Deltapmod p$. Then $0<j-ile k$ and $(j-i)n$ is equivalent mod $p$ to a unique integer in $[0,p-1]$ which is $|Delta|lefrac p{k+1}<frac p{k}$ or $p-|Delta|ge p-frac p{k+1}>p-frac p{k}$.
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
answered Feb 2 at 7:51
Alex RavskyAlex Ravsky
42.6k32383
42.6k32383
add a comment |
add a comment |
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$begingroup$
What does [a,b] mean?
$endgroup$
– William Elliot
Jan 26 at 9:45
$begingroup$
a closed interval in the real line
$endgroup$
– quantum
Jan 26 at 10:43