The Congruent Number Problem
Connection with Elliptic Curves (proof by computer)
Explicit Bijection
In fact, there is a bijection between
$$
A = \left\{(a,b,c) \in \mathbf{Q}^3 \,:\, \frac{ab}{2} = n,\, a^2 + b^2 = c^2\right\}
$$
and
$$
B = \left\{(x,y) \in \mathbf{Q}^2 \,:\, y^2 = x^3 - n^2 x, \,\,\text{with } y \neq 0\right\}
$$
given by the maps
$$
f(a,b,c) = \left(-\frac{nb}{a+c},\,\, \frac{2n^2}{a+c}\right)
$$
and
$$
g(x,y) = \left(\frac{n^2-x^2}{y},\,\,-\frac{2xn}{y},\,\, \frac{n^2+x^2}{y}\right).
$$
Define bijection between rational right triangles with area $n$ and points on the elliptic curve $y^2=x^3-n^2x$ with $y\neq 0$.
{{{id=10|
def f(a,b,c, n):
return (-n*b/(a+c), 2*n^2/(a+c))
def g(x,y, n):
return ((n^2-x^2)/y, -2*x*n/y, (n^2+x^2)/y)
///
}}}
Use computer to verify that this is a bijection.
{{{id=8|
R. = QQ[]
x,y = f(a,b,c, n)
g(x,y, n)
///
((a^2 - b^2 + 2*a*c + c^2)/(2*a + 2*c), b, (a^2 + b^2 + 2*a*c + c^2)/(2*a + 2*c))
}}}
By working in the quotient polynomial ring and avoiding fractions we get that the composition $g \circ f$ is the identity map.
{{{id=7|
Q = R.quotient([a^2 + b^2 - c^2, a*b-2*n])
v1 = g(x,y, n)
v2 = (a,b,c)
for i in range(3):
print Q(v1[i].numerator() - v2[i].numerator()*v1[i].denominator())
///
0
0
0
}}}
{{{id=15|
R. = QQ[]
a, b, c = g(x,y,n)
///
}}}
{{{id=13|
Q = R.quotient([y^2 - (x^3-n^2*x)])
v1 = f(a,b,c, n)
v2 = (x,y)
for i in range(2):
print Q(v1[i].numerator() - v2[i].numerator()*v1[i].denominator())
///
0
0
}}}
So we know that the claimed bijections are valid.
{{{id=16|
///
}}}
{{{id=5|
def cong(n):
G = EllipticCurve([-n^2,0]).gens()
if len(G) == 0: return False
x,y,_ = G[0]
return ((n^2-x^2)/y,-2*x*n/y,(n^2+x^2)/y)
///
}}}
{{{id=4|
for n in [1..15]:
print n, cong(n)
///
1 False
2 False
3 False
4 False
5 (3/2, 20/3, 41/6)
6 (3, 4, 5)
7 (-24/5, -35/12, 337/60)
8 False
9 False
10 False
11 False
12 False
13 (323/30, 780/323, 106921/9690)
14 (-8/3, -21/2, 65/6)
15 (4, 15/2, 17/2)
}}}
{{{id=21|
@interact
def _(n=6, triangles=(1..10), maxtime=(3..30)):
x,y = var('x,y')
C = EllipticCurve(y^2 == x^3 - n^2*x)
try:
alarm(maxtime)
t = walltime()
G = C.gens()
print "time = %.2f seconds"%walltime(t)
except RuntimeError:
print "Sage is unable to provably find generators"
return
except KeyboardInterrupt, msg:
print "Too hard -- timed out after %s seconds"%maxtime
return
html("rank = %s\n\n"%len(G))
if len(G) == 0: print "%s is not a congruent number"%n; return
def g(x,y,n): return ((n^2-x^2)/y, -2*x*n/y, (n^2+x^2)/y)
P = G[0]
html('Rational Right Triangles with Area %s
'%n)
for i in [1..triangles]:
a,b,c = g((i*P)[0], (i*P)[1], n)
html("(a,b,c) = $%s$\n"%latex((a,b,c)))
///
}}}
Tunnell's Criterion:
{{{id=19|
def cong_number_sets(n):
"""
Given a positive integer n, returns the two sets appearing in the
conjectural criterion for when a number is congruent.
"""
n = ZZ(n)
n = ZZ(prod([p for p, e in n.factor() if e%2 == 1]))
if n % 2 == 0: # even case
E = [] # with c even
O = [] # with c odd
a_bound = floor(sqrt(n/8) + 1)
c_bound = floor(sqrt(n)/4 + 1)
half_n = n//2
for c in range(-c_bound, c_bound+1):
c_square = c^2
for a in range(-a_bound, a_bound+1):
a_square = a^2
z = half_n - 4*a_square - 8*c_square
try:
b = z.sqrt(extend=False)
if b.denominator() == 1:
b = b.numerator()
assert 4*a^2 + b^2 + 8*c^2 == n/2
if c % 2 == 0:
E.append((a,b,c))
else:
O.append((a,b,c))
except ValueError:
pass
else:
E = [] # with c even
O = [] # with c odd
a_bound = floor(sqrt(n/2)+1)
c_bound = floor(sqrt(n/8) + 1)
for c in range(-c_bound, c_bound+1):
c_square = c^2
for a in range(-a_bound, a_bound+1):
a_square = a^2
z = n - 2*a_square - 8*c_square
try:
b = z.sqrt(extend=False)
if b.denominator() == 1:
b = b.numerator()
assert 2*a^2 + b^2 + 8*c^2 == n
if c % 2 == 0:
E.append((a,b,c))
else:
O.append((a,b,c))
except ValueError:
pass
return E, O
def is_conj_cong_number(n):
"""
Returns True if n is conjecturally a congruent number, according
to the Birch and Swinnerton-Dyer conjecture.
"""
E, O = cong_number_sets(n)
return len(E) == len(O)
///
}}}
First congruent number $\equiv 3 \pmod{8}$
{{{id=18|
time is_conj_cong_number(219)
///
True
Time: CPU 0.02 s, Wall: 0.03 s
}}}
{{{id=25|
219 % 8
///
3
}}}
{{{id=24|
cong(219)
///
(55/4, 1752/55, 7633/220)
}}}
This year isn't congruent:
{{{id=23|
time is_conj_cong_number(2010)
///
False
Time: CPU 0.02 s, Wall: 0.02 s
}}}
{{{id=3|
cong(2010)
///
False
}}}
{{{id=2|
time is_conj_cong_number(2011)
///
False
Time: CPU 0.02 s, Wall: 0.04 s
}}}
{{{id=1|
time is_conj_cong_number(2012)
///
True
Time: CPU 0.02 s, Wall: 0.02 s
}}}
{{{id=28|
cong(2012)
///
Traceback (most recent call last):
File "", line 1, in
File "_sage_input_63.py", line 9, in
exec compile(ur'open("___code___.py","w").write("# -*- coding: utf-8 -*-\n" + _support_.preparse_worksheet_cell(base64.b64decode("Y29uZygyMDEyKQ=="),globals())+"\n"); execfile(os.path.abspath("___code___.py"))' + '\n', '', 'single')
File "", line 1, in
File "/tmp/tmptRTSZC/___code___.py", line 3, in
exec compile(ur'cong(_sage_const_2012 )' + '\n', '', 'single')
File "", line 1, in
File "/tmp/tmpKsNmc0/___code___.py", line 4, in cong
G = EllipticCurve([-n**_sage_const_2 ,_sage_const_0 ]).gens()
File "/home/wstein/sage/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/ell_rational_field.py", line 1931, in gens
"\nTry increasing descent_second_limit then trying this command again."
RuntimeError: Unable to compute the rank, hence generators, with certainty (lower bound=0, generators found=[]). This could be because Sha(E/Q)[2] is nontrivial.
Try increasing descent_second_limit then trying this command again.
}}}
{{{id=29|
E = EllipticCurve([-2012^2,0])
///
}}}
{{{id=30|
E.analytic_rank()
///
1
}}}
{{{id=31|
///
}}}