Data files

This page is from 2014, but we will extend it soon with new data, probably before the end of April 2016.

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Data for rational S-unit equations

--- joint with Rafael von Känel ---

Let S be a finite set of rational primes. We consider the S-unit equation
x + y = 1,
where x and y are S-integers, i.e. rational numbers for which only primes in S appear in their numerators and denominators. To any solution (x,y) there up to are 5 further solutions, namely (y,x), (-x/y,1/y), (1/y,-x/y), (-y/x,1/x), (1/x,-y/x). Let's simply identify such pairs. If c denotes the least common denominator of x and y, we can write x=a/c and y=b/c. Therefore solving this equation is equivalent to solving
a + b = c,
where a,b,c are integers with gcd(a,b,c)=1. To break symmetry we may and do assume that 0 < a ≤ b < c.
For certain sets S, below you can download the set of all solutions (up to symmetry) to this equation.

Table 1: Solutions when S is the set of the first n primes, for 1 ≤ n ≤ 15:

S #solutions Text file (.txt) Sage file (.sobj)
2 1 .txt .sobj
2, 3 4 .txt .sobj
2, 3, 5 17 .txt .sobj
2, 3, 5, 7 63 .txt .sobj
2, 3, 5, 7, 11 190 .txt .sobj
2, 3, 5, 7, 11, 13 545 .txt .sobj
2, 3, 5, 7, 11, 13, 17 1433 .txt .sobj
2, 3, 5, 7, 11, 13, 17, 19 3649 .txt .sobj
2, 3, 5, 7, 11, 13, 17, 19, 23 8828 .txt .sobj
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 20015 .txt (0.7 MB) .sobj (0.4 MB)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 44641 .txt (1.7 MB) .sobj (0.9 MB)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 95358 .txt (3.9 MB) .sobj (1.9 MB)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 199081 .txt (10.5 MB) .sobj (4.0 MB)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 412791 .txt (22.7 MB) .sobj (8.7 MB)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 839638 .txt (47.8 MB) .sobj (18.2 MB)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 1663449 .txt (80.6 MB) .sobj (37.3 MB)

Table 2: Solutions when S is the set of the first 5 Fermat primes and 2, or the first 6 Mersenne primes and 2:

S #solutions Text file (.txt) Sage file (.sobj)
2, 3, 5, 17, 257, 65537 65 .txt .sobj
2, 3, 7, 31, 127, 8191, 131071 45 .txt .sobj

Table 3: Solutions for rad(abc) ≤ 10k, for 1 ≤ k ≤ 7:

maximal radical #solutions Text file (.txt) Sage file (.sobj)
101 5 .txt .sobj
102 42 .txt .sobj
103 354 .txt .sobj
104 2362 .txt .sobj
105 13902 .txt (0.4 MB) .sobj (0.2 MB)
106 79125 .txt (2.6 MB) .sobj (1.3 MB)
107 432408 .txt (15.0 MB) .sobj (7.3 MB)
OpenOffice Calc: In case you would like to use the data with OpenOffice Calc, simply open the .txt with this program, and at the start-up dialog add ":+=()" to "Separator options->Separated by->Others", and set "From row" to 11.

Data for the Mordell equation

--- joint with Rafael von Känel ---

Let us consider the Mordell equation
y2 = x3 + a,
over the integers with a ≠ 0. Call a solution (y,x,a) primitive if there is no real u > 1 such that (y/u3,x/u2,a/u6) is another integral solution. Using the prime factorization a = ∏pα(p), define c(a) := ∏pmin(2,α(p)). Then the file contains all primitive solutions of the above Mordell equation with c(a) ≤ 185. (More data coming soon!)
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