Contour walker

[Work in progress]

This page implements an algorithm that searches along contours of the surface v=x^3+y^3-z^3 looking for close integer approximations to v=0. The contours, which I shall label using 'n' are formed by setting z=x+n. The equation of a single contour is x^3+y^3-(x+n)^3=0. This can be rearranged to y^3-3nx^2-3n^2x-n^3=0 . This algorithm used used is a modification of Bresenham's line algorithm. This allows us to trace the contours just by adding derivatives, a lot faster than calculating y as a function of x or vice versa. This is explained in more detail on the algorithm page.

The option "Follow Contour" plots the values generated using this algorithm. We focus on the subset of contours where x>y as these approach y ∝ x^[2/3] as x gets larger. This means that the gradient approaches 1, which alllows us to accelerate the search for larger x values. This is the basis of the algorithm used in the second option, "Fast tracing". Because the gradient is close to 1 the 'v' values generated by the algorithm follow a saw-tooth pattern, so we can work of the wavelength and skip the intermediate values. This allows the algorithm to accelerate and allows me to check 100s of trillion values per day, even running javascript in a browser. See the "Algorithm details" section below.

Contour tracking

Contour: Tracking Mode: Colouring: Plot: Animate Speed:

Current: [hover mouse over grid to see values]
Latest:

 

Results

Results go here

Some Solutions

ContourDetails
1

Found 837 solutions, though I might have missed a few small ones. The last was 825=2781957377^3-2781957376^3-2852818^3

2

This generates an infinite series of solutions that add to 2. See the special cases page. The same effect can be seen on all even contours, the nth contour generates an infinite sequence of solutions for n^3/4. We are only searching for solutions up to 1024, so we only find 2→2, 4→16, 6→54, 8→128, 10→250, 12→432, 14→686 and 16→1024. To keep the output manageable the "results" section only shows the last ten examples of any repeated solution.

Ignoring the predictable solutions the furthest out I have found so far are

  • 620=8140652757^3-8140652755^3-7353429^3
  • 456=1178496770^3+2027384^3-1178496772^3

5

I ran this up to over 20 trillion, these are the only solutions where X is over a million that I found:

1101509^3-1101504^3-26304^3=701
3757419^3-3757414^3-59606^3=99
8756836^3-8756831^3-104776^3=289
4440347405^3+6662569^3-4440347410^3=134
76275664911^3+44356209^3-76275664916^3=64

7

Tested up to 1.17 trillion, solutions over 1 million (notice that 771 appears twice):

3077972^3-3077965^3-58378^3=771
14454639^3-14454632^3-163711^3=720
36460580^3-36460573^3-303358^3=771
89534194^3+552161^3-89534201^3=64
129629906^3-129629899^3-706659^3=538
9

I didn't find many solutions, I ran up to X = 129 billion, but the last solution found was for 125 at X = 196,416,133. Interestingly there are a lot of cubes, generally it seams that cubes are easier to find than other values.

1 (943  +  643  −  1033), (7293  −  7203  −  2423), (7293  +  2443  −  7383)
8 (1413  +  833  −  1503), (28403  −  28313  −  6013)
27 (463  −  373  −  363), (16523  +  4203  −  16613), (36303  −  36213  −  7083)
64 (1013  +  673  −  1103), (14473  −  14383  −  3833)
125 (1323  −  1233  −  763), (1964161333  −  1964161243  −  10136923)
216 (2583  +  1233  −  2673), (143633  +  17733  −  143723)
512 (26493  +  5753  −  26583), (140243  +  17453  −  140333)
1000 (272813  +  27193  −  272903)

11

Tested up to 14,120,117,535,495,710 (14 quadrillion), only found 3 solutions where x was over a million:

2546399^3-2546388^3-59812^3=799
7136006^3+118889^3-7136017^3=672
7265050^3+120318^3-7265061^3=451

Other pages

(c) John Whitehouse 2021 - 2023