Problems & Puzzles: Puzzles

 Puzzle 638. Special integer quotients Emmanuel Vantieghem sent the following puzzle: Everybody knows that, for any natural number  k, the expression (k+1)(k+2) ... (k+n)/(n!) is an integer.  But, what happens if we replace  n!  by the primorial  p(n)# ? Q1. Are there natural numbers  k  that make the expression (k+p(1))(k+p(2))...(k+p(n))/(p(n)#) integral ? Q2. If the answer is affirmative, are there prime  k  with that property ?

Contributions came from Jean Brette, Jan van Delden, Paul Schmidt, Farideh Firoozbakht & Jahangeer Kholdi.

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Jean wrote:

The answer to Q1 is yes.

More generally, if  S is a set of n integers  :  S= {a1, a2, ..., an}, it exists an infinite family of numbers   k  such that all the numbers  E = (k+a1) (k+a2) ...(k+an) are multiples of   P = a1*a2*....*an.

It suffice to choose  k equal a multiple of P :  k= q*P

Modulo  P , k == 0, and E became  (0+a1)(0+a2)...(0+an) == P == 0

Example :  S = {2,3,7);   k =   P    = 42 ;    E = 44*45*49 =   2310 * 42

S = {2,3,7);   k = 2* P = 84  ;    E = 86*87*91 = 16211 * 42

So, in the question Q1, and  for any  given n, choose   k = q* p(n) #, where  q is an integer. I don't know if there are other solutions.

...

One more time, I was too fast, and I did not read carefully your questions.

I think that Emmanuel is looking for some   k, independant of n,   such that the expression

E = (k+p(1)) (k+p(2)) ... (k+p(n)) / p(n) #    is an integer  for all   n.

In my last mail, I gave only an answer to a weaker question :

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Jan wrote:

Q1:         Yes.

For n fixed we would like to find k such that: p[n]#|product(k+p[i],i=1..n).

Consider any permutation of the p[i] (or a permutation s() of the indices) say p[s(i)]=q(i), we would like: p[s(i)]|(k+p[i]) for every i.

Or in other words: p[i]=-k mod p[s(i)].

Due to the Chinese Remainder Theorem there is always a solution for every permutation s().

A simple (not the smallest!) solution would be: k=p[n]# itself (or 0, but that's not a natural number), where s() is just the identity operator, since p[i] is a divisor of p[n]#.

Finding the smallest solutions for every n is a lot of work since there are n! different permutations s().

However!. Finding the smallest solution using this technique is no garantee that this generates the overall smallest solution, since we could have p[s(j)]|(k+p[i]) for j<>i as well.

smallest solutions using the permutation technique, n=1..10:

[2,1,3,2,4,19,15,1579,21,6]

smallest solutions using a search on k:

[2,1,3,2,4,8,15,63,21,6]

Q2:         Yes.

From Q1 we know that we have solutions k=a mod p[n]#, for n! different a (applying the Chinese Remainder Theorem).

If we can prove that among those a there is at least one for which (a,p[n]#)=1, we are done upon using Dirichlet's theorem. (*)

[Note this includes the prime solutions q, with q>p[n], a situation that appears to include all possible prime solutions if n>4 (**)].

The number of permutations s(), generating a residu a with (a,p[n]#)=1 can be calculated using round(n!/e) exactly and is therefore >=1 for all n.

(**)

If we have a solution a=p[i] we should have p[j]|(a+p[k]).

If a=2 it's easy to see that the last three primes should have a difference of 2, hence n=4.

If a>2, k>1, we have a+p[k] even. We would like p[n] to be a divisor, giving a=p[k]=p[n].    (Only situation: n=4?)

If a>2, k=1, we have a+p[k] odd. We would like p[n] to be a divisor, giving a=p[n]-2=p[n-1]. (Only situation: n=3,4?)

smallest prime solution using the permutation technique:

[2,7,3,2,19,19,193,1579,9059,6203]

smallest prime solution using a search on k:

[2,3,3,2,17,19,193,401,401,401]

(*)

The permutations s() of 1..n, which have at least one index i=i* which remains unchanged, are called matching permutations. For such an index i* we get k=-p[i*] mod p[i*] or k=0 mod p[i*], hence p[i*]|(k,p[n]#) or (k,p[n]#)>1.

The permutations s() of 1..n, which have no index i with s(i)=i, are called derangements. It is known how many there are given n, and this will result in (k,p[n]#)=1, since all congruences are not 0 mod p[i].

The number of derangements given n are: n!*sum((-1)^j/j!,j=0..n). This can be proven by using the inclusion/exclusion principle, or simply put, by correctly counting.

If one looks at the taylor polynom for exp(x) =sum(1/j!*x^j, j=0..inf) and substitute -1 we get:

exp(-1)=sum((-1)^j/j!,j=0..inf) thus:

The number of derangements given n are: n!*[The value of exp(-1) cut down at term n].

The reason that this number of derangements can be computed using the round function:

round(n! exp(-1))= round( number of derangements + n!*rest series) = number of derangements + round(n! * rest series), with:

n!*rest series =n!*sum((-1)^j/j!,j=n+1..inf)  and this should be absolutely bounded by 0.5 in order for the round() function to return 0.

Any standard calculus book will tell us it’s absolutely bounded by 1/(n+1) and hence by 0.5 if n>=1.

Note: the used property of the primes is (p[i],p[j])=1. So more general: product((k+a[i])/a[i],i=1..n) integer, will have a prime solution k if for all i,j with i<>j we have (a[i],a[j])=1.

***

Paul wrote:

Q1. k = a*p(n)# where ‘a’ is all natural numbers
Q2. No, there are no prime k

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Farideh & Jahangeer wrote:

Yes for each natural number n, there exist infinitely many numbers k such
that  f(k, n) = (k+p(1)) (k+p(2))...(k+p(n))/(p(n)#) is integer,
Prove : Take k = m*p(n)#, m is any natural number. f(m*(p(n)#), n) is integer
because if  i <=  n then, p(i) divides  m*(p(n)#) + p(i).

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If a(n) be the smallest k such that f(k, n) is integer, then a(n) for n = 1, 2, ..., 25 are:
2, 1, 3, 2, 4, 8, 15, 63, 21, 6, 167, 55, 684, 2191, 45, 1961, 13671, 3202, 43470,13828, 161725, 161725, 1104891, 1327050 & 1327050.

Example : a(25) = 1327050 ==> f(1327050, 25) is integer,
f(1327050, 25) = 512567564042748032834583551993336383696
41475188081351051755691243483237458583499359462427260608
4927921126872917782286.

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Answer to Q2 :  We think that for each natural number n, there exist
primes k such that,  f(k, n) = (k+p(1))(k+p(2))...(k+p(n)) / (p(n)#) is integer.

If b(n) be the smallest prime k, such that f(k, n) is integer, then b(n)
for n = 1, 2, ..., 25 are: for n = 1, 2, ... , 25  are :
2, 3, 3, 2, 17, 19, 193, 401, 401, 401, 167, 2477, 6871, 12119, 121119,  21383,
104281, 104281, 152213, 152213, 175919, 1786159, 7255159, 7035733&
154272581.

Example : b(25) = 154272581 ==> f(154272581, 25) is integer,
f(154272581) = 22102616568537022572958847825630504990540
874543400821315309199684373371770620920957469850607484
636881871070759547080646971662632819662111592257154330
81788202116710400000.

***

This is the solution by Emmanuel, who posed this puzzle:

We will write the n-th prime as  p(n) ; the product of the first  n
primes will be denoted by  P(n) and  the product  (x+p(1))...(x+p(n))
will be abbreviated to  P(x,n).

The first question is trivial : for instance,  k = P(n)-p(i)  makes
the quotient integral for every  i = 1, 2, ... , n.

Next, I prove the following proposition :
for  n > 1, there exists an integer  q, relatively prime to  P(n)
such that  P(q,n)  is divisible by  P(n).
We prove this by induction.
The proposition is true for  n = 2.  Indeed, P(1,2) = (1+2)(1+3)  is
divisible by  P(2) = 6.
Assume the proposition is true for some  n > 2.  Thus, there exists
some  k, relatively prime to  P(n)  such that  P(k,n)  is divisble by
P(n).  Clearly, for every value of  m, the number  q = k+m P(n)  is
also relatively prime to  P(n)  and  P(q,n)  is divisible by  P(n).
Now, we can choose  m  such that  q+p(1) = q+2  is divisible by
p(n+1).  This implies that  q  is  relatively prime to  P(n+1)  and
that  P(q,n)  is divisble by  P(n+1).  Hence,  P(q,n+1) =
P(q,n)*(q+p(n+1))  is divisble by  P(n+1), which proves the
proposition for  n+1.

Now, we can answer the second question.
When  n = 1, there is one and only one prime  q  that makes  (q+2)/2
integral : q= 2.
For  n > 1,  if  q  is relatively prime to  P(n)  and such that
P(q,n)/P(n)  is integral, we can replace  q  by  q+m P(n).  By
Dirichlet's theorem on primes in arithmetic progressions there are
infinitely many  m  such that  q+mP(n)  is prime.
_____________________________________

Off the record, I made a small table that gives for  n = 2, 3, ...
the smallest prime  q  for which  P(q,n)/P(n)  is integral :
n = 2 ; q = 3
n = 3 ; q = 3
n = 4 ; q = 2
n = 5 ; q = 17
n = 6 ; q = 19
n = 7 ; q = 193
n = 8 ; q = 401
n = 9 ; q = 401
n=10 ; q = 401
n = 11 ; q = 167
n = 12 ; q = 2477
n = 13 ; q = 6871
n = 14 ; q = 12119
n = 15 ; q = 12119
n = 16 ; q = 21383
n = 17 ; q = 104281
n = 18 ; q = 104281
n = 19 ; q = 152213
n = 20 ; q = 152213
n = 21 ; q = 175919
n = 22 ; q = 1786159
n = 23 ; q = 7255159
n = 24 ; q = 7035733
n = 25 ; q = 154272581

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