33 as a sum of three cubes - NARS2000

[brent.hildebrand]

A new math result recently found:

(x y z)←8866128975287528x ¯8778405442862239x ¯2736111468807040x
      +/x y z * 3
33

OK - it is just interesting to me... 

[Davin Church]

Cute!  But them's big numbers!

[Bob Smith]

Me, too.  Where did you read of this?

BTW, you need only one suffix of 'x' in a strand to mark the entire strand as Multiple-Precision.  Typically it's easiest to suffix just the rightmost number.  Also, you can use a suffix of 'x' on floating point numbers, too, as in

      3.1x
31r10

[brent.hildebrand]

Here is the article describing the problem and technique used to solve this sum or three cubes problem:

https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf

Here is a Numberphile video interview on the finding:
https://youtu.be/ASoz_NuIvP0

[brent.hildebrand]

Here is just running the result in the language Julia which has many APLish ideas.  Julia has a REPL (read, execute, print and loop) giving it the interactive feel of the APL session.  It also can show you the code that it emits when it create a function.  Again - it is just interesting to me...

julia> f(x) = sum(x.^3)

f (generic function with 1 method)

julia> x = (8866128975287528, -8778405442862239, -2736111468807040)

(8866128975287528, -8778405442862239, -2736111468807040)

julia> f(x)

33

julia> @code_native f(x)

.text

; ┌ @ REPL[1]:1 within `f'

; │┌ @ broadcast.jl:753 within `materialize'

; ││┌ @ broadcast.jl:953 within `copy'

; │││┌ @ tuple.jl:161 within `ntuple'

; ││││┌ @ broadcast.jl:953 within `#19'

; │││││┌ @ broadcast.jl:551 within `_broadcast_getindex'

; ││││││┌ @ broadcast.jl:578 within `_broadcast_getindex_evalf'

; │││││││┌ @ intfuncs.jl:244 within `literal_pow'

; ││││││││┌ @ operators.jl:502 within `*' @ REPL[1]:1

movq (%rdi), %rax

movq %rax, %rcx

imulq %rcx, %rcx

imulq %rax, %rcx

vmovdqu 8(%rdi), %xmm0

vpmullq %xmm0, %xmm0, %xmm1

vpmullq %xmm0, %xmm1, %xmm0

; │└└└└└└└└

; │┌ @ tuple.jl:395 within `sum'

; ││┌ @ operators.jl:502 within `+' @ int.jl:53

vmovq %xmm0, %rdx

addq %rcx, %rdx

vpextrq $1, %xmm0, %rax

addq %rdx, %rax

; │└└

retq

nopw %cs:(%rax,%rax)

; └

[brent.hildebrand]

Here is one place that NARS2000 needs several 'x's to get the right answer:

      *(163x*.5x)×○1
262537412640768333.516358125973357125101
      *(163x*.5x)×○1x
262537412640768743.999999999999250072589
      *(163x*.5)×○1x
262537412640768743.999999999999250072589
      *(163*.5)×○1x
262537412640768247.571355416771299107802
      *(163x*.5)×○1x
262537412640768743.999999999999250072589  <---  correct answer.  x is needed on 163 and ○1x.  Probably due to right to left interpretation? 

It is enteresting that e raised to the power of pi times the squareroot of 163 is so close to a real number....   .999999999999  close.   :-)