``Congruent number curves'' ny²=x³-x with |n| congruent to 5 mod 8 and less than a million: Positivity of rank, and list of curves of conjectural rank 3

For which numbers n is there a rational Pythagorean triangle of area n? This problem is known to be equivalent to: for which n does the elliptic curve

have a rational point other than the point at infinity and the 2-torsion points with y=0? Such n are said to be ``congruent'' because they are also known to be the numbers that occur as the common difference (``congruum'' in Latin) of a three-term arithmetic progression of squares.

We may, and henceforth do, assume that n is a positive squarefree integer. It is conjectured that if n is congruent mod 8 to 5, 6, or 7, then n is congruent. See this page, concerning the 8k+7 case, for more background on this problem and the relevant conjectures.

Theorem. If n=8k+5 and n<10⁶ then n is a congruent number.

The proof, combining computation and theory, proceeds as described in the 8k+7 page, except that the Heegner-point construction is replaced by a variant, called ``mock Heegner points'' by Monsky [see his Math. Zeitschrift paper ``Mock Heegner points and congruent numbers'' (Vol.204 (1990) #1, 45-67) -- though Bryan Birch notes that this variant is if anything closer to Heegner's original construction...]. As a result, we cannot use Gross-Zagier to deduce that the analytic rank of E(n) is at least 3 if the resulting point P(n) is torsion, and 1 if not. Nor can we use Kolyvagin to deduce that the arithmetic rank is 1 in the non-torsion case. But it seems a reasonable conjecture that both Gross-Zagier and Kolyvagin extend to the ``mock Heegner'' context. At any rate, to complete the proof of the theorem it suffices to exhibit a single rational non-torsion point on each of the 381 curves E(n) on which P(n) is numerically indistinguishable from a torsion point. We do this in the table at the end of this page, giving for each n the numerator and denominator of the x-coordinate of a nontrivial rational solution of ny²=x³-x. The points with large numerator and denominator were obtained with Cremona's MWRANK.

In all 381 cases, 2-descent on E(n) and/or on a curve 2-isogenous with E(n) shows that the arithmetic rank is at most 3. In all but 18 cases, the 2-part of the Tate-Shafarevich group SHA of E(n) is trivial if E(n) has arithmetic rank 3 as expected. The 18 exceptional cases are:

n = 80189 = 17*53*89	n = 82205 = 5*41*401	n = 83845 = 5*41*409	n = 92045 = 5*41*449
n = 116645 = 5*41*569	n = 125749 = 13*17*569	n = 182005 = 5*89*409	n = 199805 = 5*89*449
n = 270805 = 5*41*1321	n = 282613 = 41*61*113	n = 342205 = 5*89*769	n = 440453 = 13*17*1993
n = 477581 = 13*17*2161	n = 517789 = 41*73*173	n = 689005 = 5*41*3361	n = 749989 = 17*157*281
n = 758693 = 13*17*3433	n = 982085 = 5*13*29*521

In each of these cases, MWRANK confirms arithmetic rank 3 (by 2-descent on an isogenous curve), but finds a subgroup of SHA isomorphic with (Z/2Z)².