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A104461
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Number of instances of nonprimes m in Pythagorean triples x,y,z such that x^2 + y^2 = z^2. Except for 1, the number of instances of composite numbers m in Pythagorean triples.
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0
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0, 1, 1, 2, 2, 2, 4, 1, 5, 3, 2, 5, 4, 1, 7, 4, 2, 3, 4, 5, 4, 4, 2, 5, 7, 1, 5, 8, 4, 4, 8, 1, 10, 2, 4, 5, 5, 3, 5, 7, 4, 2, 14, 1, 7, 5, 8, 4, 5, 4, 5, 12, 2, 9, 4, 4, 5, 11, 4, 2, 13, 8, 1, 5, 7, 8, 5, 4, 4, 1, 5, 13, 2, 7, 9, 5, 8, 14, 2, 10, 5, 5, 10, 4, 5, 5, 8, 1, 5, 23, 2, 2, 5, 4, 6, 7, 6, 4, 8, 13
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OFFSET
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1,4
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COMMENTS
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The PARI script is direct and very fast for m = x,y values but slows in the trial routine for m = z. We save some time for m even allowing the testing of only even values of y.
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LINKS
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FORMULA
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Consider Pythagorean triples x^2 + y^2 = z^2. We seek to find the total number of instances of an integer m being x or y or z. The solution for x or y is straightforward by considering appropriate lesser and greater pairwise factors, L, G of m^2 in z^2 - y^2 = (z-y)(z+y) = m^2. Then solve for z and y with the relations, z-y = L z+y = G 2z = L+G, z = (L+G)/2 where L and G are both even if m is even or both odd if m is odd. The number of L factors < m is the number of instances of x or y. The count of instances z=m is solved by trial on x^2 = m^2 - y^2.
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EXAMPLE
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For m=30 there are 5 Pythagorean triples that have a 30:
30, 224, 226
30, 72, 78
30, 40, 50
30, 16, 34
18, 24, 30
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PROG
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(PARI)
\\ instances of m in Pythagorean triples using a direct method for x, y
pythm3(m) = { local(m2, ln, j, j2=0, d, d2, q2, q, a, b, x, x1, x2, xx, y, y2, z, c, c2, r, f, str, stp); d=divisors(m^2); /* get the divisors of m^2 */ ln=length(d)-1; d2=q2=vector(ln); m2=m^2; if(m%2, r=1, r=0); for(j=1, ln, /* save only the both even r=0, both odd r=1 */ if(d[j]%2==r, if(m2/d[j]%2==r, j2++; d2[j2]=d[j]; q2[j2]=m2/d[j]; /* save m/factor to solve (z-y)(z+y) = m^2 */ ) ) ); x2=y2 = vector(20); for(j=1, j2, z=(d2[j] + q2[j])/2; y= z - d2[j]; if(y>0, c++; ) ); if(m%2==0, start=2; step=2, start=1; step=1); forstep(y=start, m-1, step, /* esolve when z is m */ x1 = (m2-y^2); if(issquare(x1), c2++; x2[c2]=floor(sqrt(x1)); /* save to later mask dupes */ y2[c2]=y; ) ); for(x=1, c2, /* mask the dupes routine */ for(y=x, c2, if(x2[x]==y2[y], ) ) ); return(c+c2/2) /* print total */}
for(k=1, 400, if(isprime(k)==0, print1(pythm3(k)", ")))
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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