Oops! I had made a MAPLE script to illustrate the proof in action & verify it. The script worked, and the proof was verified... but... when MAPLE fully simplified the stuff in the script, it turned out 2 of my 4 parameters in my "4-parameter family" of Heron tetrahedra... vanish! They cancel themselves out! So it is really only a 2-parameter family and my power law like N^0.4999 is really going to be a considerably weaker power law, I must retract and correct that: REVISED CONCLUSION: We have found a class of Heron tetrahedra (actually several classes, but just the first will suffice) with sides s01=a, s23=b, s02=s03=s12=s13=c such that the number of primitive tetrahedra in this class with max(a,b,c)<N grows with N at least as the same order as N^(1/14). ----- #Here is a MAPLE script which computes members of my #infinite 4-param family of Heron tetrahedra and verifies their Heronitude: #Warren D. Smith, Dec 2011 aa := (n,m) -> 16*m*n*(n^2+m^2)*(n^4+m^4)*(n^4+14*m^2*n^2+m^4)* (n^4+6*m^2*n^2+m^4)*(n^2-4*m*n-m^2)* (n^2+4*m*n-m^2)*(17*n^8+28*n^6*m^2+166*n^4*m^4+28*n^2*m^6+17*m^8); bb := (n,m) -> 6*(n^2+m^2)*(n^8-4*n^6*m^2+134*n^4*m^4-4*n^2*m^6+m^8)* (5*n^4+6*m^2*n^2+5*m^4)*(n^8+60*n^6*m^2+134*n^4*m^4+60*n^2*m^6+m^8)*(m-n)^3*(m+n)^3; cc := (n,m) -> (n^8-4*n^6*m^2+134*n^4*m^4-4*n^2*m^6+m^8)*(m^4+n^4)* (n^8+60*n^6*m^2+134*n^4*m^4+60*n^2*m^6+m^8)*(17*n^8+28*n^6*m^2+166*n^4*m^4+28*n^2*m^6+17*m^8); print("the following (a,b,c,area1,area2,vol) tuples should be all-integer:"); for n from 1 to 9 do for m from n to 9 do a := aa(n,m): b := bb(n,m): c := cc(n,m): g := gcd(a, gcd(b,c)): a := abs(a/g): b := abs(b/g): c := abs(c/g): if(a<>0 and b<>0 and c<>0) then print( "(", a, b, c, #sides sqrt(a^2*(2*c+a)*(2*c-a)/16), #triarea sqrt(b^2*(2*c+b)*(2*c-b)/16), #triarea sqrt( a^2*b^2*(4*c^2-a^2-b^2)/144 ), #tetvol ")" ); fi;od;od;