I’ve seen a number of question asking for solutions of $kx^4+1=y^2$ where $k$ is a small fixed value, and found these frustrating when I find just two, or fewer solutions.
Just for fun, I wondered what happened when I fixed $x$ to investigate what values of $k$ were produced. There are far more than I presumed, and (after rather more effort than I expected), some fit into parametric solutions.
In the following $a$ and $b>2$ are positive integers and the pairs of $\pm$ are both plus or both minus.
$$(k,x,y)=(a^2-1,1,a)$$
$$(k,x,y)=(a(2^4a\pm2),2,2^4a\pm1)$$
$$(k,x,y)=(a(2^6a\pm1),4,2^7a\pm1)$$
$$(k,x,y)=(a(2^{4b-2}a\pm1),2^b,2^{4b-1}a\pm1)$$
Examples for $(k,x,y)=(a^2-1,1,a)$ with $a=2,3,4,5,6,7$
$$(3,1,2)$$ $$(8,1,3)$$ $$(15,1,4)$$ $$(24,1,5)$$ $$(35,1,6)$$ $$(48,1,7)$$
Examples for $(k,x,y)=(a(2^4a\pm2),2,2^4a\pm1)$ with $a=1,2,3$, minus then plus.
$$(14,2,15)$$ $$(60,2,31)$$ $$(138,2,47)$$ $$(18,2,17)$$ $$(68,2,33)$$ $$(150,2,49)$$
Examples for $(k,x,y)=(a(2^6a\pm1),4,2^5a\pm1)$ with $a=1,2,3$, minus then plus.
$$(63,4,127)$$ $$(254,4,255)$$ $$(573,4,383)$$ $$(65,4,129)$$ $$(258,4,257)$$ $$(579,4,385)$$
Examples for $(k,x,y)=(a(2^{4b-2}a\pm1),2^b,2^{4b-1}a\pm1)$ with $b=3,a=1,2,3$, minus then plus.
$$(1023,8,2047)$$ $$(4094,8,4095)$$ $$(9213,8,6143)$$ $$(1025,8,2049)$$ $$(4098,8,4097)$$ $$(9219,8,6145)$$
Examples for $(k,x,y)=(a(2^{4b-2}a\pm1),2^b,2^{4b-1}a\pm1)$ with $b=4,a=1,2,3$, minus then plus.
$$(16383,16,32767)$$ $$(65534,16,65535)$$ $$(147453,16,98303)$$ $$(16385,16,32769)$$ $$(65538,16,65537)$$ $$(147459,16,98305)$$
Examples for $(k,x,y)=(a(2^{4b-2}a\pm1),2^b,2^{4b-1}a\pm1)$ with $b=5,a=1,2,3$, minus then plus.
$$(262143,32,524287)$$ $$(1048574,32,1048575)$$ $$(2359293,32,1572863)$$ $$(262145,32,524289)$$ $$(1048578,32,1048577)$$ $$(2359299,32,1572865)$$
Please accept my advance apologies for typos.
Updates 6 June 2017
Thanks to a comment from @JyrkiLahtonen I’ve found, and corrected, a program bug that incorrectly gave solutions for some high values. Hence, I’ve removed the restriction that $k$ not be square, as it’s unnecessary.
My question:
Please could you find more parametric solutions to $kx^4+1=y^2$ ? I can almost detect patterns, but just can’t resolve them.
This is only an attempt at a partial answer.
I’ve found a solution that gives, IMHO, an infinite set of results for every positive value of $x$.
$$(k,x,y)=(a^2x^4\pm2a,x,ax^4\pm1)$$
where $a$ is a positive integers and the pairs of $\pm$ are both plus or both minus, except for $x=1$ where both are plus only.
However, this does not cover all solutions, except perhaps for $x=1$.
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I’ve also found solutions for individual values of $x$, but feel these are of limited value or interest.
For example, with $x=2$, we have, $$(k,x,y)=( 16a^2-18a+5,2,16a-9)$$ $$(k,x,y)=( 16a^2-14a+3,2,16a-7)$$
Taken with $$(k,x,y)=(a(2^4a\pm2),2,2^4a\pm1)$$
these cover all the solutions for $x=2$ that I’ve found.