Help21:=proc(): print(`  Tn(n,A,B,C) , IsC , JTseq(N), g(q,N) `): end:



#Tn(n,A,B,C): The set of (x,y,z) such that n=A*x^2+B*y^2+C*z^2
Tn:=proc(n,A,B,C) local x,y,z,S:

S:={}:
for z from 0 to trunc(sqrt(n/C)) do
 for x from  0 to trunc((n-C*z^2)/2) do
  y:=sqrt((n-C*z^2-A*x^2)/B):

   if  type(y,integer) then
     S:=S union {[x,y,z]}:
   fi:
 od:
od:
S:
end:


#IsC(n): is n a congruent number?
IsC:=proc(n)
 if n mod 2=1 then
   evalb(nops(2*Tn(n,2,1,32))= nops(Tn(n,2,1,8))):
else
   evalb(nops(2*Tn(n,8,2,64))= nops(Tn(n,8,2,16))):
fi:
end:

#JTseq(N): all the congruent numbers <=N
JTseq:=proc(N) local n,L:
L:=[]:

for n from 1 to N do
 if IsC(n) then
  L:=[op(L),n]:
 fi:
od:

L:

end:

#g(q,N): The truncated g in Junnell's paper
g:=proc(q,N) local n,f:

f:=q*mul((1-q^(8*n))*(1-q^(16*n)),n=1..trunc(N/8)):

taylor(f,q=0,N+1):

end: