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Looking for a square result (Posted on 2018-06-14) Difficulty: 3 of 5
For each positive integer n, let Mn be the square matrix (nxn) where each diagonal entry is 2018, and every other entry is 1.

Determine the smallest positive integer n (if any) for which the value
of det(Mn) is a perfect square.

No Solution Yet Submitted by Ady TZIDON    
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re(2): computer solution | Comment 5 of 18 |
(In reply to re: computer solution by Charlie)

Right - I assumed only the main diagonal was 2018s  - I wonder which way is right. 


I also encountered roundoff errors. Going to higher precision I get 
possibly that n=6, 8, 10 may be squares. But I think the polynomial form should be able to be expressed as a square, no? 


        program dee

        implicit none

        real*8 xn,xi,xj,det,d,rdet

        integer n

        d=2018.

        print*,'         n              det             sqrt(det)'

        print*,'         ----------------------------------------'


          do n=3,10

          xn=n

                if(2*(n/2).eq.n)then

                xi=xn/2.

                det= d**xn - xi*d +(xi-1)       

                else

                det= d**xn - xn*d + (xn-1) 

                endif

          rdet=sqrt(det)

          print*,n,det,rdet

          enddo

        end


          n              det             sqrt(det)

          ----------------------------------------

           3   8217943780.0000000        90652.875188821228     

           4   16583822756941.000        4072323.9995045825     

           5   33466154331639484.        182937569.49199769     

           6   6.7534699441268818E+019   8217949831.9999990     

           7   1.3628502347248049E+023   369168015234.90692     

           8   2.7502317736746564E+026   16583822760976.000     

           9   5.5499677192754564E+029   744981054744042.25     

          10   1.1199834857497871E+033   33466154331649568.     

Edited on June 15, 2018, 9:37 am
  Posted by Steven Lord on 2018-06-15 09:35:39

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