## Clone product

**Moderators:** gmalivuk, Moderators General, Prelates

### Clone product

Hi,

Let x(1),x(2),x(3).....x(k) be integers such as :

x(1) < x(2) < x(3).....< x(k-1) < x(k)

Let r be an integer positive number > 0

Show that :

(x(k+1))^2 + r = product ((x(i)^2+r) with i varying from 1 to k

Let x(1),x(2),x(3).....x(k) be integers such as :

x(1) < x(2) < x(3).....< x(k-1) < x(k)

Let r be an integer positive number > 0

Show that :

(x(k+1))^2 + r = product ((x(i)^2+r) with i varying from 1 to k

- Forest Goose
**Posts:**377**Joined:**Sat May 18, 2013 9:27 am UTC

### Re: Clone product

Are you asking if given an increasing sequence, there always exists such r? The nature of x given an r? If it holds for any x and any r? It's not clear what you're looking for.

Forest Goose: A rare, but wily, form of goose; best known for dropping on unsuspecting hikers, from trees, to steal sweets.

### Re: Clone product

r is a constant number

it always exists a set of x(i) values making the equation hold.

it always exists a set of x(i) values making the equation hold.

### Re: Clone product

r is a constant number

it always exists a set of x(i) values making the equation hold.

All the factors have the same form (x(i)^2+r) and the product too

it always exists a set of x(i) values making the equation hold.

All the factors have the same form (x(i)^2+r) and the product too

### Re: Clone product

As a short example

r=1

170=17*5*2

x(i)={1,2,4}

the product = (13^2)+1

For any fixed r we could generate an infinite number of infinite sequences x(i)

r=1

170=17*5*2

x(i)={1,2,4}

the product = (13^2)+1

For any fixed r we could generate an infinite number of infinite sequences x(i)

### Re: Clone product

What I think the OP's problem is:

Fix r > 0. Given a sequence a(1) < a(2) < ... < a(k), does the quantity (a(1)^2 + r)(a(2)^2 + r)...(a(k)^2 + r) have the form a(k+1)^2 +r for a(k+1) > a(k)?

Fix r > 0. Given a sequence a(1) < a(2) < ... < a(k), does the quantity (a(1)^2 + r)(a(2)^2 + r)...(a(k)^2 + r) have the form a(k+1)^2 +r for a(k+1) > a(k)?

**Spoiler:**

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### Re: Clone product

Correct.

Just prove that :

((x^2+r)*((x+1)^2+r))-r is perfect square then you could build an infinite number of infinite sequences fullfilling such equality.

Anyway that it not the only way to build other sequences.

Than you.

Just prove that :

((x^2+r)*((x+1)^2+r))-r is perfect square then you could build an infinite number of infinite sequences fullfilling such equality.

Anyway that it not the only way to build other sequences.

Than you.

### Re: Clone product

Here is a short example where a difference is bigger than 1.

r=3

x(1)=8

x(2)=91

x(3)=745

(8^2+3)*(91^2+3)=745^2+3

67*8284=555028

We could find longer sequences with different gaps.

r=3

x(1)=8

x(2)=91

x(3)=745

(8^2+3)*(91^2+3)=745^2+3

67*8284=555028

We could find longer sequences with different gaps.

### Re: Clone product

Goahead52 wrote:Correct.

Just prove that :

((x^2+r)*((x+1)^2+r))-r is perfect square then you could build an infinite number of infinite sequences fullfilling such equality.

((x+1)

^{2}+r)*(x

^{2}+r) = (x

^{2}+x+r)

^{2}+ r

### Re: Clone product

Goahead52 wrote:Hi,

Let x(1),x(2),x(3).....x(k) be integers such as :

x(1) < x(2) < x(3).....< x(k-1) < x(k)

Let r be an integer positive number > 0

Show that :

(x(k+1))^2 + r = product ((x(i)^2+r) with i varying from 1 to k

Instead of r can we find a general solution with -r (all the other conditions remain identical)

Thanks a lot

### Re: Clone product

jaap wrote:Goahead52 wrote:Correct.

Just prove that :

((x^2+r)*((x+1)^2+r))-r is perfect square then you could build an infinite number of infinite sequences fullfilling such equality.

((x+1)^{2}+r)*(x^{2}+r) = (x^{2}+x+r)^{2}+ r

Thanks a lot

### Re: Clone product

Hi,

How can we link those forms (x^2+ r) or (x^2-r) I mean the same r but with different x`s with factorials and/or combinatorics identities?

In fact the goal is to express some factorial n using those forms but in a way such as the formula will be elegant.

Good luck!

How can we link those forms (x^2+ r) or (x^2-r) I mean the same r but with different x`s with factorials and/or combinatorics identities?

In fact the goal is to express some factorial n using those forms but in a way such as the formula will be elegant.

Good luck!

### Re: Clone product

As an example :

13!=(63^2+74879)*(64^2+74879)

r=74879

x(1)=63

x(2)=64

13!=(63^2+74879)*(64^2+74879)

r=74879

x(1)=63

x(2)=64

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