$x^2-Dy^2=-1$ when D is square-free and has only prime divisors $\equiv 1 \mod 4$

Question

I wish to know whether the Pell equation $x^2-Dy^2=-1$ can have any solutions when $D>0$ is a square-free product of primes $\equiv 1 \mod 4$ only. If there are any possible D for which this is the case, then I wish to know whether any of these D can be of the form $k^2-16$, where $k$ is an odd positive integer $\equiv 1 \mod 4$, hence a product of primes $\equiv 1 \mod 4$ and square of primes $\equiv 3 \mod 4$. https://ac.els-cdn.com/S0022314X0500123X/1-s2.0-S0022314X0500123X-main.pdf?_tid=f71361cb-b5e4-4806-8e25-93a86e5cd716&acdnat=1526891273_4f3552881e5d1d486a04658e936f66e7 Seems to prsent an answer but I fail to see whether D leads to on odd graph or not. So I am looking for an easier explanation.

Note that $k-4$ and $k+4$ do not have prime factors 3, so 9 divides $k$.

Answer 1

The article can be downloaded from here.. Things begin to make some sense: the author, Chun-Gang Ji, thanks Fei Xu at the end. Fei Xu went to Germany (not sure of the years) on a Humboldt grant and studied with Rainer Schulze-Pillot.

alright, I found that your conditions allow for $ D =81 w^2 - 16$ to have factors of $7, 11, 19,$ or other primes $q \equiv 3 \pmod 4.$ This immediately rules out any integer $x^2 - D y^2 = -1.$

I thought about it, I am not at all sure what numbers you want. I did a new run, where I just took $w^2 - 16$ when it is divisible only by primes $p \equiv 1 \pmod 4$

Mon May 21 19:13:05 PDT 2018
      9          65 = 5 * 13                8^2 - 65 * 1^2 = -1 
     21         425 = 5^2 * 17              268^2 - 425 * 13^2 = -1 
     33        1073 = 29 * 37               45368^2 - 1073 * 1385^2 = -1
     57        3233 = 53 * 61               1065376^2 - 3233 * 18737^2 = -1 
     69        4745 = 5 * 13 * 73           173932^2 - 4745 * 2525^2 = -1
     93        8633 = 89 * 97         !!!!! NNNOOOOOOOOOO!!!!!!!!!!!!!!!!!!             
    105       11009 = 101 * 109             164922396062746432^2 - 11009 * 1571830457668225^2 = -1 
    141       19865 = 5 * 29 * 137          95780490068^2 - 19865 * 679567765^2 = -1 
    153       23393 = 149 * 157             20407772838045547232^2 - 23393 * 133429743458877905^2 = -1 
    177       31313 = 173 * 181             6345269649362901313958144318096^2 - 31313 * 35858138775030566830162906553^2 = -1
    189       35705 = 5 * 37 * 193          911752230148^2 - 35705 * 4825166629^2 = -1 
    201       40385 = 5 * 41 * 197    !!!!! NNNOOOOOOOOOO!!!!!!!!!!!!!!!!!!       
    225       50609 = 13 * 17 * 229         316331912320^2 - 50609 * 1406141833^2 = -1 
    237       56153 = 233 * 241       !!!!! NNNOOOOOOOOOO!!!!!!!!!!!!!!!!!! 
    261       68105 = 5 * 53 * 257
    273       74513 = 269 * 277
    285       81209 = 17^2 * 281
    309       95465 = 5 * 61 * 313
    321      103025 = 5^2 * 13 * 317
    369      136145 = 5 * 73 * 373
    393      154433 = 389 * 397
    405      164009 = 401 * 409
    429      184025 = 5^2 * 17 * 433
    453      205193 = 449 * 457
    489      239105 = 5 * 17 * 29 * 97
    537      288353 = 13 * 41 * 541
    561      314705 = 5 * 113 * 557
    573      328313 = 569 * 577
    597      356393 = 593 * 601
    621      385625 = 5^4 * 617
    657      431633 = 653 * 661
    681      463745 = 5 * 137 * 677
    693      480233 = 13 * 17 * 41 * 53
    705      497009 = 701 * 709
    729      531425 = 5^2 * 29 * 733
    765      585209 = 761 * 769
    789      622505 = 5 * 13 * 61 * 157
    825      680609 = 821 * 829
    849      720785 = 5 * 13^2 * 853
    861      741305 = 5 * 173 * 857
    933      870473 = 929 * 937
    945      893009 = 13 * 73 * 941
    981      962345 = 5 * 197 * 977
   1017     1034273 = 1013 * 1021
   1029     1058825 = 5^2 * 41 * 1033
   1065     1134209 = 1061 * 1069
   1101     1212185 = 5 * 13 * 17 * 1097
   1113     1238753 = 1109 * 1117

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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ date
Mon May 21 19:32:21 PDT 2018
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell   8633


0  form   1 184 -169   delta  -1
1  form   -169 154 16   delta  10
2  form   16 166 -109   delta  -1
3  form   -109 52 73   delta  1
4  form   73 94 -88   delta  -1
5  form   -88 82 79   delta  1
6  form   79 76 -91   delta  -1
7  form   -91 106 64   delta  2
8  form   64 150 -47   delta  -3
9  form   -47 132 91   delta  1
10  form   91 50 -88   delta  -1
11  form   -88 126 53   delta  2
12  form   53 86 -128   delta  -1
13  form   -128 170 11   delta  16
14  form   11 182 -32   delta  -5
15  form   -32 138 121   delta  1
16  form   121 104 -49   delta  -2
17  form   -49 92 133   delta  1
18  form   133 174 -8   delta  -22
19  form   -8 178 89   delta  2
20  form   89 178 -8   delta  -22
21  form   -8 174 133   delta  1
22  form   133 92 -49   delta  -2
23  form   -49 104 121   delta  1
24  form   121 138 -32   delta  -5
25  form   -32 182 11   delta  16
26  form   11 170 -128   delta  -1
27  form   -128 86 53   delta  2
28  form   53 126 -88   delta  -1
29  form   -88 50 91   delta  1
30  form   91 132 -47   delta  -3
31  form   -47 150 64   delta  2
32  form   64 106 -91   delta  -1
33  form   -91 76 79   delta  1
34  form   79 82 -88   delta  -1
35  form   -88 94 73   delta  1
36  form   73 52 -109   delta  -1
37  form   -109 166 16   delta  10
38  form   16 154 -169   delta  -1
39  form   -169 184 1   delta  184
40  form   1 184 -169

 disc 34532
Automorph, written on right of Gram matrix:  
6789127433661761  1255404293921176880
7428427774681520  1373619837975061441


 Pell automorph 
690204482704361601  64129616978825562160
7428427774681520  690204482704361601

Pell unit 
690204482704361601^2 - 8633 * 7428427774681520^2 = 1 

=========================================

8633      89 *  97

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ date
Mon May 21 19:32:28 PDT 2018

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Answer 2

If $p \equiv 1 \pmod 4$ is a (positive) prime number, then there is a solution to $$ x^2 - p y^2 = -1 $$

If $p,q \equiv 1 \pmod 4$ are (positive) primes with Legendre symbol $$ (p|q)= (q|p) = -1, $$ then there is a solution to $$ x^2 - pq y^2 = -1. $$ If, instead, $ (p|q)= (q|p) = 1, $ there may or not be solutions. For example there are no solutions to $$ x^2 - 205 y^2 = -1 $$ or $$ x^2 - 221 y^2 = -1$$ But there are solutions to $x^2 - 145 y^2 = -1$

Answer 3

I have now checked $x^2 - D y^2 = -1$ with $D = 81 w^2 - 16$ and $w \equiv 1 \pmod 4.$ Seems a mixed bag so far, there are integer solutions for $$ w = 1, 17, 21, 25, 29, 41, 45, ... $$ but not for $$ w = 5, 9, 13, 33, 37, 49, ... $$

The lengths of the continued fractions have not stabilized, which means that this needs to be split up into several cases, at best, at worst has no pattern discernible without class field theory.

The question becomes: why do you care about this?