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