Divisibility of prime factorial
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If $p>3$ is a prime number and $q$ is the next prime following $p$, is it always true that $p!$ is divisible by all $n<q$?
elementary-number-theory prime-numbers
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up vote
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favorite
If $p>3$ is a prime number and $q$ is the next prime following $p$, is it always true that $p!$ is divisible by all $n<q$?
elementary-number-theory prime-numbers
add a comment |
up vote
6
down vote
favorite
up vote
6
down vote
favorite
If $p>3$ is a prime number and $q$ is the next prime following $p$, is it always true that $p!$ is divisible by all $n<q$?
elementary-number-theory prime-numbers
If $p>3$ is a prime number and $q$ is the next prime following $p$, is it always true that $p!$ is divisible by all $n<q$?
elementary-number-theory prime-numbers
elementary-number-theory prime-numbers
asked Nov 15 at 15:59
Alex
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604
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2 Answers
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up vote
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The answer is that $n$ divides $p!$ if $p<n<q$.
Let $p<n<q$ and assume that $r$ is a prime and $r^kmid n$. Our goal is to prove that $r^kmid p!$. Assume the contrary. First, note that in fact, we see that $r^k=n$. Indeed: if $n>r^k$ then $nge 2r^k$. On the other hand, Bertrand's postulate implies that $q<2p$ and we have
$$r^klefrac n2<frac q2<p$$
and then $r^kmid p!$.
Moreover, $k>1$. Otherwise, $r^k$ would be a prime, a contradiction with the fact that $q$ is the least prime after $p$.
Now we have
$$k>sum_{j=1}^inftyleftlfloorfrac p{r^j}rightrfloor=sum_{j=1}^{k-1}leftlfloorfrac p{r^j}rightrfloorgesum_{j=1}^{k-1}leftlfloorfrac {r^{k-1}}{r^j}rightrfloor=sum_{j=0}^{k-2}r^jge1+2(k-2)=2k-3$$
That is, $k<3$. This implies $k=2$.
So we have now three primes $p,q,r$ such that $p<r^2<q$ such that $lfloor p/rrfloor<2$, that is, $p<2r$, and then $r^2<q<2p<4r$, so $r<4$. This leaves us with two options: $n=4$ and $n=9$. But $5le p<n$ and the last prime before $9$ is $7$, and $9mid 7!$.
1
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
|
show 6 more comments
up vote
1
down vote
Obviously $p!$ is divisible by any $n leq p$. So we need only concern ourselves with $p < n < q$.
As you have stipulated that $q$ is the very next prime greater than $p$, it follows that the numbers between $p$ and $q$ are composite numbers divisible only by primes less than or equal to $p$.
So $gcd(n, p!) geq 2$. Here's where your question gets interesting: it might be possible that one of these $n$ is divisible by some prime $r < p$ but with a higher exponent than in $p!$.
The likeliest candidate for such a prime is 2. Since $p$ is odd, $p!$ has $$2^frac{p - 1}{2}$$ as a divisor.
But let's not forget that multiples of 4 contribute at least twice as much as singly even numbers to 2's exponent in $p!$. So, assuming $p equiv 1 bmod 4$, the larger number $$2^{frac{p - 1}{2} + frac{p - 1}{4}}$$ is also a divisor of $p!$ (just a small tweak if $p equiv 3 bmod 4$ instead).
So the best chance to accumulate enough exponents of 2 would be with the very first prime greater than a given power of 2, let's say $2^m$. But no even number between $2^m$ and $2^{m + 1}$ has a higher exponent for 2 than $m$.
I guess this is the point where we must invoke Bertrand's postulate... or maybe it's the prime number theorem that we need instead? The fact $$frac{2^m}{log 2^m} < frac{2^{m + 1}}{log 2^{m + 1}}$$ by more than 2 for $m > 4$ suggests that there are at least two primes between $2^m$ and $2^{m + 1}$.
Okay, so what if instead we choose the prime right below $2^m$? But thanks to 2 and $2^{m - 1}$, $p!$ has at least $m$ for 2's exponent in its factorization.
Surely there are some gaps in the reasoning above, but I hope at least you find it illuminating.
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
The answer is that $n$ divides $p!$ if $p<n<q$.
Let $p<n<q$ and assume that $r$ is a prime and $r^kmid n$. Our goal is to prove that $r^kmid p!$. Assume the contrary. First, note that in fact, we see that $r^k=n$. Indeed: if $n>r^k$ then $nge 2r^k$. On the other hand, Bertrand's postulate implies that $q<2p$ and we have
$$r^klefrac n2<frac q2<p$$
and then $r^kmid p!$.
Moreover, $k>1$. Otherwise, $r^k$ would be a prime, a contradiction with the fact that $q$ is the least prime after $p$.
Now we have
$$k>sum_{j=1}^inftyleftlfloorfrac p{r^j}rightrfloor=sum_{j=1}^{k-1}leftlfloorfrac p{r^j}rightrfloorgesum_{j=1}^{k-1}leftlfloorfrac {r^{k-1}}{r^j}rightrfloor=sum_{j=0}^{k-2}r^jge1+2(k-2)=2k-3$$
That is, $k<3$. This implies $k=2$.
So we have now three primes $p,q,r$ such that $p<r^2<q$ such that $lfloor p/rrfloor<2$, that is, $p<2r$, and then $r^2<q<2p<4r$, so $r<4$. This leaves us with two options: $n=4$ and $n=9$. But $5le p<n$ and the last prime before $9$ is $7$, and $9mid 7!$.
1
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
|
show 6 more comments
up vote
3
down vote
accepted
The answer is that $n$ divides $p!$ if $p<n<q$.
Let $p<n<q$ and assume that $r$ is a prime and $r^kmid n$. Our goal is to prove that $r^kmid p!$. Assume the contrary. First, note that in fact, we see that $r^k=n$. Indeed: if $n>r^k$ then $nge 2r^k$. On the other hand, Bertrand's postulate implies that $q<2p$ and we have
$$r^klefrac n2<frac q2<p$$
and then $r^kmid p!$.
Moreover, $k>1$. Otherwise, $r^k$ would be a prime, a contradiction with the fact that $q$ is the least prime after $p$.
Now we have
$$k>sum_{j=1}^inftyleftlfloorfrac p{r^j}rightrfloor=sum_{j=1}^{k-1}leftlfloorfrac p{r^j}rightrfloorgesum_{j=1}^{k-1}leftlfloorfrac {r^{k-1}}{r^j}rightrfloor=sum_{j=0}^{k-2}r^jge1+2(k-2)=2k-3$$
That is, $k<3$. This implies $k=2$.
So we have now three primes $p,q,r$ such that $p<r^2<q$ such that $lfloor p/rrfloor<2$, that is, $p<2r$, and then $r^2<q<2p<4r$, so $r<4$. This leaves us with two options: $n=4$ and $n=9$. But $5le p<n$ and the last prime before $9$ is $7$, and $9mid 7!$.
1
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
|
show 6 more comments
up vote
3
down vote
accepted
up vote
3
down vote
accepted
The answer is that $n$ divides $p!$ if $p<n<q$.
Let $p<n<q$ and assume that $r$ is a prime and $r^kmid n$. Our goal is to prove that $r^kmid p!$. Assume the contrary. First, note that in fact, we see that $r^k=n$. Indeed: if $n>r^k$ then $nge 2r^k$. On the other hand, Bertrand's postulate implies that $q<2p$ and we have
$$r^klefrac n2<frac q2<p$$
and then $r^kmid p!$.
Moreover, $k>1$. Otherwise, $r^k$ would be a prime, a contradiction with the fact that $q$ is the least prime after $p$.
Now we have
$$k>sum_{j=1}^inftyleftlfloorfrac p{r^j}rightrfloor=sum_{j=1}^{k-1}leftlfloorfrac p{r^j}rightrfloorgesum_{j=1}^{k-1}leftlfloorfrac {r^{k-1}}{r^j}rightrfloor=sum_{j=0}^{k-2}r^jge1+2(k-2)=2k-3$$
That is, $k<3$. This implies $k=2$.
So we have now three primes $p,q,r$ such that $p<r^2<q$ such that $lfloor p/rrfloor<2$, that is, $p<2r$, and then $r^2<q<2p<4r$, so $r<4$. This leaves us with two options: $n=4$ and $n=9$. But $5le p<n$ and the last prime before $9$ is $7$, and $9mid 7!$.
The answer is that $n$ divides $p!$ if $p<n<q$.
Let $p<n<q$ and assume that $r$ is a prime and $r^kmid n$. Our goal is to prove that $r^kmid p!$. Assume the contrary. First, note that in fact, we see that $r^k=n$. Indeed: if $n>r^k$ then $nge 2r^k$. On the other hand, Bertrand's postulate implies that $q<2p$ and we have
$$r^klefrac n2<frac q2<p$$
and then $r^kmid p!$.
Moreover, $k>1$. Otherwise, $r^k$ would be a prime, a contradiction with the fact that $q$ is the least prime after $p$.
Now we have
$$k>sum_{j=1}^inftyleftlfloorfrac p{r^j}rightrfloor=sum_{j=1}^{k-1}leftlfloorfrac p{r^j}rightrfloorgesum_{j=1}^{k-1}leftlfloorfrac {r^{k-1}}{r^j}rightrfloor=sum_{j=0}^{k-2}r^jge1+2(k-2)=2k-3$$
That is, $k<3$. This implies $k=2$.
So we have now three primes $p,q,r$ such that $p<r^2<q$ such that $lfloor p/rrfloor<2$, that is, $p<2r$, and then $r^2<q<2p<4r$, so $r<4$. This leaves us with two options: $n=4$ and $n=9$. But $5le p<n$ and the last prime before $9$ is $7$, and $9mid 7!$.
edited Nov 15 at 19:32
answered Nov 15 at 17:25
ajotatxe
52.1k23688
52.1k23688
1
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
|
show 6 more comments
1
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
1
1
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I'm wondering if there is a short proof without Bertrand's postulate.
– ajotatxe
Nov 15 at 17:37
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
I know it might be a stupid question, but could you tell me what does $r^kmid n$ mean in the first line ?
– Sauhard Sharma
Nov 15 at 17:45
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
$mid $ means 'evenly divides' as $2mid 6$. $r^k mid n$ means that $n=ccdot r^k$ where $c$ is an integer.
– Keith Backman
Nov 15 at 17:50
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@ajotatxe I'm not clear on how you can limit the exponent of $r$ to $2$. The question as posed can be seen to ask as a particular cases whether $128mid 127!$ or $27mid 23!$. But $128ne r^2$ and $27ne r^2$ for any integer $r$. Other examples of this kind are easy to generate by taking an odd power of a prime ($u=s^{2t+1}$) and asking if $u$ divides the factorial of the largest prime smaller than $u$.
– Keith Backman
Nov 15 at 18:22
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
@KeithBackman It is proved in my answer that the exponent must be lesser than $3$.
– ajotatxe
Nov 15 at 19:02
|
show 6 more comments
up vote
1
down vote
Obviously $p!$ is divisible by any $n leq p$. So we need only concern ourselves with $p < n < q$.
As you have stipulated that $q$ is the very next prime greater than $p$, it follows that the numbers between $p$ and $q$ are composite numbers divisible only by primes less than or equal to $p$.
So $gcd(n, p!) geq 2$. Here's where your question gets interesting: it might be possible that one of these $n$ is divisible by some prime $r < p$ but with a higher exponent than in $p!$.
The likeliest candidate for such a prime is 2. Since $p$ is odd, $p!$ has $$2^frac{p - 1}{2}$$ as a divisor.
But let's not forget that multiples of 4 contribute at least twice as much as singly even numbers to 2's exponent in $p!$. So, assuming $p equiv 1 bmod 4$, the larger number $$2^{frac{p - 1}{2} + frac{p - 1}{4}}$$ is also a divisor of $p!$ (just a small tweak if $p equiv 3 bmod 4$ instead).
So the best chance to accumulate enough exponents of 2 would be with the very first prime greater than a given power of 2, let's say $2^m$. But no even number between $2^m$ and $2^{m + 1}$ has a higher exponent for 2 than $m$.
I guess this is the point where we must invoke Bertrand's postulate... or maybe it's the prime number theorem that we need instead? The fact $$frac{2^m}{log 2^m} < frac{2^{m + 1}}{log 2^{m + 1}}$$ by more than 2 for $m > 4$ suggests that there are at least two primes between $2^m$ and $2^{m + 1}$.
Okay, so what if instead we choose the prime right below $2^m$? But thanks to 2 and $2^{m - 1}$, $p!$ has at least $m$ for 2's exponent in its factorization.
Surely there are some gaps in the reasoning above, but I hope at least you find it illuminating.
add a comment |
up vote
1
down vote
Obviously $p!$ is divisible by any $n leq p$. So we need only concern ourselves with $p < n < q$.
As you have stipulated that $q$ is the very next prime greater than $p$, it follows that the numbers between $p$ and $q$ are composite numbers divisible only by primes less than or equal to $p$.
So $gcd(n, p!) geq 2$. Here's where your question gets interesting: it might be possible that one of these $n$ is divisible by some prime $r < p$ but with a higher exponent than in $p!$.
The likeliest candidate for such a prime is 2. Since $p$ is odd, $p!$ has $$2^frac{p - 1}{2}$$ as a divisor.
But let's not forget that multiples of 4 contribute at least twice as much as singly even numbers to 2's exponent in $p!$. So, assuming $p equiv 1 bmod 4$, the larger number $$2^{frac{p - 1}{2} + frac{p - 1}{4}}$$ is also a divisor of $p!$ (just a small tweak if $p equiv 3 bmod 4$ instead).
So the best chance to accumulate enough exponents of 2 would be with the very first prime greater than a given power of 2, let's say $2^m$. But no even number between $2^m$ and $2^{m + 1}$ has a higher exponent for 2 than $m$.
I guess this is the point where we must invoke Bertrand's postulate... or maybe it's the prime number theorem that we need instead? The fact $$frac{2^m}{log 2^m} < frac{2^{m + 1}}{log 2^{m + 1}}$$ by more than 2 for $m > 4$ suggests that there are at least two primes between $2^m$ and $2^{m + 1}$.
Okay, so what if instead we choose the prime right below $2^m$? But thanks to 2 and $2^{m - 1}$, $p!$ has at least $m$ for 2's exponent in its factorization.
Surely there are some gaps in the reasoning above, but I hope at least you find it illuminating.
add a comment |
up vote
1
down vote
up vote
1
down vote
Obviously $p!$ is divisible by any $n leq p$. So we need only concern ourselves with $p < n < q$.
As you have stipulated that $q$ is the very next prime greater than $p$, it follows that the numbers between $p$ and $q$ are composite numbers divisible only by primes less than or equal to $p$.
So $gcd(n, p!) geq 2$. Here's where your question gets interesting: it might be possible that one of these $n$ is divisible by some prime $r < p$ but with a higher exponent than in $p!$.
The likeliest candidate for such a prime is 2. Since $p$ is odd, $p!$ has $$2^frac{p - 1}{2}$$ as a divisor.
But let's not forget that multiples of 4 contribute at least twice as much as singly even numbers to 2's exponent in $p!$. So, assuming $p equiv 1 bmod 4$, the larger number $$2^{frac{p - 1}{2} + frac{p - 1}{4}}$$ is also a divisor of $p!$ (just a small tweak if $p equiv 3 bmod 4$ instead).
So the best chance to accumulate enough exponents of 2 would be with the very first prime greater than a given power of 2, let's say $2^m$. But no even number between $2^m$ and $2^{m + 1}$ has a higher exponent for 2 than $m$.
I guess this is the point where we must invoke Bertrand's postulate... or maybe it's the prime number theorem that we need instead? The fact $$frac{2^m}{log 2^m} < frac{2^{m + 1}}{log 2^{m + 1}}$$ by more than 2 for $m > 4$ suggests that there are at least two primes between $2^m$ and $2^{m + 1}$.
Okay, so what if instead we choose the prime right below $2^m$? But thanks to 2 and $2^{m - 1}$, $p!$ has at least $m$ for 2's exponent in its factorization.
Surely there are some gaps in the reasoning above, but I hope at least you find it illuminating.
Obviously $p!$ is divisible by any $n leq p$. So we need only concern ourselves with $p < n < q$.
As you have stipulated that $q$ is the very next prime greater than $p$, it follows that the numbers between $p$ and $q$ are composite numbers divisible only by primes less than or equal to $p$.
So $gcd(n, p!) geq 2$. Here's where your question gets interesting: it might be possible that one of these $n$ is divisible by some prime $r < p$ but with a higher exponent than in $p!$.
The likeliest candidate for such a prime is 2. Since $p$ is odd, $p!$ has $$2^frac{p - 1}{2}$$ as a divisor.
But let's not forget that multiples of 4 contribute at least twice as much as singly even numbers to 2's exponent in $p!$. So, assuming $p equiv 1 bmod 4$, the larger number $$2^{frac{p - 1}{2} + frac{p - 1}{4}}$$ is also a divisor of $p!$ (just a small tweak if $p equiv 3 bmod 4$ instead).
So the best chance to accumulate enough exponents of 2 would be with the very first prime greater than a given power of 2, let's say $2^m$. But no even number between $2^m$ and $2^{m + 1}$ has a higher exponent for 2 than $m$.
I guess this is the point where we must invoke Bertrand's postulate... or maybe it's the prime number theorem that we need instead? The fact $$frac{2^m}{log 2^m} < frac{2^{m + 1}}{log 2^{m + 1}}$$ by more than 2 for $m > 4$ suggests that there are at least two primes between $2^m$ and $2^{m + 1}$.
Okay, so what if instead we choose the prime right below $2^m$? But thanks to 2 and $2^{m - 1}$, $p!$ has at least $m$ for 2's exponent in its factorization.
Surely there are some gaps in the reasoning above, but I hope at least you find it illuminating.
answered Nov 16 at 17:16
Robert Soupe
10.6k21949
10.6k21949
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