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  • 15 Comments
Joined 1 year ago
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Cake day: June 14th, 2023

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  • I did find that it can be done arbitrarily. Mind is definitely not into writing about it, though, but here’s the gp code I wrote to look it over.

    /*
        There may exist a 0<=t<s such that
        s divides both x and (x+(x%d)*(t*d-1))/d.
    
    
        To show this for solving for divisibility of 7 in 
        any natural number x.
    
        g(35,5,10) = 28
        g(28,5,10) = 42
        g(42,5,10) = 14
        g(14,5,10) = 21
        g(21,5,10) =  7
    */
    
    g(x,t,d)=(x+(x%d)*(t*d-1))/d;
    
    /* Find_t( x = Any natural number that is divisible by s,
               s = The divisor the search is being done for,
               d = The modulus restriction ).
    
        Returns all possible t values.
    */
    
    Find_t(x,s, d) = {
        V=List();
        
        for(t=2,d-1,
            C = factor(g(x,t,d));
            for(i=1,matsize(C)[1],if(C[i,1]==s, listput(V,t))));
            
        return(V);
    }   
    

    One thing that I noticed almost right away, regardless what d is, it seems to always work when s is prime, but not when s is composite.

    Too tired…Pains too much…Have to stop…But still…interesting.