Re: S-Unit questions

[Kurt Foster <drsardonicus@earthlink.net>]

On Mar 14, 2026, at 4:38 AM, hermann@stamm-wilbrandt.de wrote:

> Below SageMath solver run for S={2,3} took less than 5 minutes.
> I tried with S={2,3,5,7} and aborted after 22.5h on AMD 7950X CPU  
> with boost frequency 5.4GHz run.
> https://gist.github.com/Hermann-SW/b538c991c922b1e8fc06abbab554b3ef?permalink_comment_id=6027812 
> #gistcomment-6027812
> Then I tried with S={2,3,5} and aborted after 13.5h.

To find small solutions to u + v = 1 in rational fractions  with u, v  
S-units, I tried the following approach: First, generate a list of  
positive integers up to a limit (here, 1000)` that are factorable over  
S. Then find relatively prime pairs whose differences are also  
factorable over S. These lead to 3 formally different sets of  
solutions; only [2,-1] occurs more than once. I choose N as a product  
of powers of primes in S, so that each prime-power factor is as big as  
my size limit. Of course, there is no guarantee of finding all  
solutions.

For S = {2, 3, 5} I obtained the following in short order:

? N=2^10*3^7*5^5;v=divisors(N);
? w=[];for(i=1,#v,if(v[i]<=1000,w=concat(w,[v[i]]),break))
? for(i=1,#w-1,for(j=i+1,#w,if(gcd(w[i],w[j])==1,d=w[j]-w[i];if(N 
%d==0,s1=[w[i]/w[j],d/w[j]];s2=[w[j]/w[i],-d/w[i]];s3=[w[j]/d,-w[i]/ 
d];print(s1", "s2", "s3)))))
[1/2, 1/2], [2, -1], [2, -1]
[1/3, 2/3], [3, -2], [3/2, -1/2]
[1/4, 3/4], [4, -3], [4/3, -1/3]
[1/5, 4/5], [5, -4], [5/4, -1/4]
[1/6, 5/6], [6, -5], [6/5, -1/5]
[1/9, 8/9], [9, -8], [9/8, -1/8]
[1/10, 9/10], [10, -9], [10/9, -1/9]
[1/16, 15/16], [16, -15], [16/15, -1/15]
[1/25, 24/25], [25, -24], [25/24, -1/24]
[1/81, 80/81], [81, -80], [81/80, -1/80]
[2/3, 1/3], [3/2, -1/2], [3, -2]
[2/5, 3/5], [5/2, -3/2], [5/3, -2/3]
[2/27, 25/27], [27/2, -25/2], [27/25, -2/25]
[3/4, 1/4], [4/3, -1/3], [4, -3]
[3/5, 2/5], [5/3, -2/3], [5/2, -3/2]
[3/8, 5/8], [8/3, -5/3], [8/5, -3/5]
[3/128, 125/128], [128/3, -125/3], [128/125, -3/125]
[4/5, 1/5], [5/4, -1/4], [5, -4]
[4/9, 5/9], [9/4, -5/4], [9/5, -4/5]
[5/6, 1/6], [6/5, -1/5], [6, -5]
[5/8, 3/8], [8/5, -3/5], [8/3, -5/3]
[5/9, 4/9], [9/5, -4/5], [9/4, -5/4]
[5/32, 27/32], [32/5, -27/5], [32/27, -5/27]
[8/9, 1/9], [9/8, -1/8], [9, -8]
[9/10, 1/10], [10/9, -1/9], [10, -9]
[9/25, 16/25], [25/9, -16/9], [25/16, -9/16]
[15/16, 1/16], [16/15, -1/15], [16, -15]
[16/25, 9/25], [25/16, -9/16], [25/9, -16/9]
[24/25, 1/25], [25/24, -1/24], [25, -24]
[25/27, 2/27], [27/25, -2/25], [27/2, -25/2]
[27/32, 5/32], [32/27, -5/27], [32/5, -27/5]
[80/81, 1/81], [81/80, -1/80], [81, -80]
[125/128, 3/128], [128/125, -3/125], [128/3, -125/3]

I then started over with {2,3,5} taking N=2^13*3^9*5^6 and a limit of  
10,000. No new solutions turned up.

I tried S = {2,3,5,7}. There were quite a few solutions, with a few  
more showing up when I raised the limit from 1000 to 10,000.