[seqfan] Re: Triangles of sums

hv at crypt.org hv at crypt.org
Mon Jul 26 18:23:53 CEST 2021


I wrote:
:Allan Wechsler <acwacw at gmail.com> wrote:
::Returning to Recamán's original suggestion (entries must be prime or
::square), I wonder if anyone can quickly show a height-3 difference triangle
::with all square entries, or prove it impossible?
:
:This was the smallest I found, testing only selected products of distinct
:primes of the form 4k+1 as the square root of the apex:
:    142129
:  67600 74529
:4096 63504 11025
:
:(Roots: 377; 260, 273; 64, 252, 105.)
:
:I imagine height-4 and above are also possible, but likely to be rather
:harder to find. Maybe I'll give it a go: establishing least apex for
:heights 3 and 4 would be enough to justify a sequence, I think, given
:height 1 is degenerate and 2 is trivial.

Oh, actually much easier than I had anticipated:

 25
9 16

    625
  225 400
81 144 256

      15625
    5625 10000
  2025 3600 6400
729 1296 2304 4096

        390625
      140625 250000
    50625 90000 160000
  18225 32400 57600 102400
6561 11664 20736 36864 65536

etc.

Hugo


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