If we have distinct primes $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = -1,$ there is a solution to $u^2 - pq v^2 = -1$ in integers and the fundamental unit of $O_{\mathbb Q(\sqrt{pq})} $ has norm $-1.$ Stevenhagen attributes this to Dirichlet (1834).
There is no such result for $p \equiv q \equiv 1 \pmod 4,$ with Legendre $(p|q) = (q|p) = 1.$ I did a census, out of the first 300,000 such numbers, there were 99284 for which $u^2 - pq v^2 = -1$ was possible, or very close to one out of three. Here are the first 100 such, and the first 200 not:
145 445 901 1145 1313 1745 2249 2305 2501 2545
2605 2705 3029 3161 3341 3545 3601 3845 4045 4777
5045 5245 5305 5545 5629 5713 5933 6145 6245 6401
6445 6649 6757 6893 6953 6989 7045 7093 7397 7745
7837 7897 8005 8077 8345 8545 8653 8945 9089 9305
9673 9881 9953 10001 10081 10145 10345 10405 10445 10777
10817 10961 11029 11141 11237 11453 11629 11729 11945 12181
12389 12461 12629 12773 12961 13105 13169 13549 13645 13801
14305 14933 15245 15397 15445 15509 15529 15845 15929 15949
16153 16601 16609 16645 16801 16837 16861 16945 17305 17345
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205 221 305 377 505 545 689 745 793 905
1205 1345 1405 1469 1513 1517 1537 1717 1945 1961
2005 2041 2045 2105 2245 2329 2353 2533 2669 2701
2845 2993 3005 3205 3305 3497 3505 3737 3805 3893
4069 4105 4145 4321 4369 4381 4405 4453 4633 4645
4705 4717 4849 4981 5017 5057 5069 5105 5141 5249
5345 5513 5645 5809 5905 5917 5989 6001 6005 6341
6497 6505 6605 6613 6697 6773 6805 6905 7081 7145
7157 7289 7361 7405 7445 7453 7769 7813 7957 8033
8045 8105 8149 8333 8357 8473 8489 8605 8621 8633
8705 8749 8801 9005 9077 9113 9445 9469 9505 9701
9745 9797 9809 9841 10121 10237 10361 10421 10441 10517
10573 10645 10705 10805 10841 10877 11105 11345 11357 11405
11513 11545 11705 11905 11917 12017 12137 12205 12317 12469
12605 12745 12869 12913 12937 13045 13073 13141 13213 13253
13445 13637 13705 13745 13817 13837 13897 13945 13969 13973
14005 14093 14209 14257 14309 14417 14473 14521 14545 14689
14701 14761 14845 14893 14989 15005 15109 15205 15293 15305
15545 15605 15677 15769 15857 15905 16021 16045 16105 16145
16201 16237 16409 16441 16505 16769 16805 16913 17113 17197
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Note that you can quickly check the above at http://www.numbertheory.org/php/unit.html
Question: in the limit, do exactly one third of such $n=pq,$ with $p \equiv q \equiv 1 \pmod 4,$ and Legendre $(p|q) = (q|p) = 1,$ allow $u^2 - pq v^2 = -1?$ Or, more to the point I suppose, $x^2 + xy - \frac{pq-1}{4} y^2 = -1?$