Squares and sequence

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Goahead52
Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

Squares and sequence

Postby Goahead52 » Sun Nov 01, 2015 3:00 pm UTC

Hi everyone,

Let us take the sequence 1,2,3,4,5,6,7,8

By using the product of pairs we could produce a square :

8*7+6*5+4*3+2*1=100=10^2

In this case in particular all the product are between consecutive numbers.

The puzzle is :

Producing squares under constraints :
- The sequence used have to start from 1 to 2n
- You can multiply only pairs
- You have to use each number only once
- All the numbers of the sequence must be used.
- The sum of all the products by pair must be equal to some square number

n=1
With 1,2 you can not produce a square 1*2=2 is not square
n=2
With 1,2,3,4
1*2+3*4=15 not square
1*3+2*4=11 not square
1*4+2*3=10 not square

and so on

n=1 and n=3 have no solutions
n=4 have a solution k=10^2

Can you find the value of the first n producing 2 squares or more?

curiosityspoon
Posts: 35
Joined: Wed Sep 24, 2014 5:01 pm UTC

Re: Squares and sequence

Postby curiosityspoon » Mon Nov 02, 2015 7:39 am UTC

1x8 + 2x7 + 3x4 + 5x6 already gets you 64, and 1x7 + 2x3 + 4x5 + 6x8 gets you 81.

Goahead52
Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

Re: Squares and sequence

Postby Goahead52 » Mon Nov 02, 2015 9:47 am UTC

curiosityspoon wrote:1x8 + 2x7 + 3x4 + 5x6 already gets you 64, and 1x7 + 2x3 + 4x5 + 6x8 gets you 81.

Thank you.
Correct!
n=4
There are 3 solutions : 8^2,9^2,10^2

Did you find it using a program or by hand?

Goahead52
Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

Re: Squares and sequence

Postby Goahead52 » Mon Nov 02, 2015 2:09 pm UTC

This puzzle could lead to a conjecture :

For n>0 any sequence 1,2,3,4,....2n will generate under the constraints (see above) a set of values containing at least one prime number.

n=1
2 is prime
n=2
11 is prime
n=3
29,31,37,41 are all primes

and so on

I want your thoughts.

As counterexample can you find a value n of where the set of values generated contains only composite numbers?

Thank you for any clue.

Goahead52
Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

Re: Squares and sequence

Postby Goahead52 » Wed Nov 04, 2015 2:42 pm UTC

Here I computed some values

Column 1 : 2n
Column 2 : minimal value
Column 3 : maximal value

2n min max
2 2 2
4 10 14
6 28 44
8 60 100
10 110 190
12 182 322
14 280 504
16 408 744
18 570 1050
20 770 1430
22 1012 1892
24 1300 2444
26 1638 3094
28 2030 3850
30 2480 4720

I need your help to know what is the asymptotic limit of max/min
I`m not sure that the ratio max/min goes to infinite when n grows
I want to know if for n>=4 all the squares between min and max have always solution.

Thank you.

Goahead52
Posts: 431
Joined: Thu Oct 16, 2014 9:28 am UTC

Re: Squares and sequence

Postby Goahead52 » Wed Nov 04, 2015 5:52 pm UTC

The limit of the ratio max/min is 2 when n goes to infinite.
As all the values between max and min are not covered (maybe half of them) if there is no range between min and max composed only with composite we could say that the range is thiner than the one of Bertrand postulate. If we remove the even numbers from that range it will give us a cardinal of that set around 1/4 of 2n.

I do not know if you see what I`m pointing out to.
Ps : Sorry I`m not english native spoken.


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