Find all $m \geq 0$ such that there are no positive integers $x,y$ that satisfy $32xy+10y-3x-1 = 2^m$

Question

Is there any efficient approach to answer this question and similar ones?

for $m=0,1,2,3,4,5$ it's easy to see there are no solutions, because the left hand side is at least $32\cdot 1 \cdot 1 + 10 \cdot 1 - 3 \cdot 1 - 1 = 38$. but what about the rest? is there some sophisticated approach to take care of the cases for which $2^m$ is larger than the lower bound mentioned above?

Thanks

Answer 1

There are plenty of solutions : The solutions upto $m=60$ are :

? for(j=0,60,n=2^(j+4)+1;fordiv(n,s,t=n/s;x=(s-5)/16;y=(t+3)/32;if(x>0,if(y>0,if
(denominator(x)==1,if(denominator(y)==1,print([x,y,j])))))))
[35, 1, 10]
[3, 39569, 22]
[26715, 5, 22]
[33, 62954, 26]
[95, 22003, 26]
[1073, 1954, 26]
[16925, 124, 26]
[1100145, 2, 26]
[225339, 149, 30]
[35, 243254785, 38]
[89, 96178414, 38]
[4515, 1902401, 38]
[5805, 1479668, 38]
[1463385, 5870, 38]
[145823823, 59, 38]
[187119065, 46, 38]
[9478548515, 1, 38]
[17, 7938712114, 42]
[63, 2170802819, 42]
[3134733, 43844, 42]
[145298705, 946, 42]
[530844479, 259, 42]
[16781, 131040004, 46]
[6392195, 344017, 46]
[17200845, 127844, 46]
[347457075, 6329, 46]
[27868809575, 79, 46]
[562949953421, 4, 46]
[3028041, 2974595894, 58]
?

For $m=86$ and $m=98$ , many pairs $(x/y)$ do the job.

Probably for infinite many $m$ , there are solutions (Seems that $m$ must be of the form $4k+2$), but I think it is very difficult to find all such $m's$.