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While taking a brute-force look at this question I discovered that it seems that almost every prime (I'll conjecture every prime larger than 20627) can be written as $p=w^2+wc+d$ for $w,c,d\in \mathbb{Z}^+$ and with $cd$ a perfect cube.

This can be rearranged to say that for almost any $p$ there is a positive integer $a<p$ such that the equation $$ y^2=x^3+\frac{1}{4}+\frac{p}{a^2} $$ has a solution of the form $(x,y)=(-u/a,v/(2a))$ where $u,v\in \mathbb{Z}^+$. (This is slightly weaker, to yield the form for $p$ above we also need $a\mid u^3$ and $v\equiv a\pmod{2}$.)

This looks like a variation of seeking integer points on a Mordell curve, but allowing solutions on a finer grid. It might be a more difficult way of looking at the problem, but does it seem like some theory of elliptic curves could help classify when there's a solution (in general, or say given $a=1$ or $a=2$)? Or, given that there is a solution for $a=1$, does this suggest an algorithm for finding it?

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Why didn't you start looking at the homogenous spaces and then dig for points for given $a$ ?. More over Nagell-Lutz theorem would be helpful in finding out the torsion part. So the best way to see is to take Selmer and Sha and perform a 2-descent. The ingredients include : 1) Rational points on elliptic curves by Silverman ( Ask me if you don't have a copy with you ) 2) this article 3) this question.. Contd. –  Iyengar Jun 24 '12 at 5:04
    
But as far as I know there is no such promising algorithm that will always gives you points. If the Birch and Swinnerton-Dyer conjecture is true, then the coefficient predicted can be of some help for seeking the points ( Given if the curve has rank $r=1$ ) but apart from that there are no other methods known. Perhaps if you don't get an answer here, you can try posting the same at MO where people like W.Stein and B.J.Poonen, Silverman can help you. They are big shots in this field. ;-) . My best wishes. –  Iyengar Jun 24 '12 at 5:07
    
Thanks @Iyengar, I will consult those references. –  Zander Jun 24 '12 at 13:18
    
For a gentle introduction into the subject why can't you try reading the lecture notes by Prof.Alvaro Lozano Robledo at PCMI, where he explicitly and very neatly describe the points computation and clear construction of Selmer and Tate-Shafarevich group and 2-descent. Thats one of the most beautiful article I ever read. But start computing yourself. Because, many people left me helping regarding that subject. I then remembered the Einstein's Quote : " I am thankful to all those who said "NO" to me when I asked something, its because of them I did it myself ". So give it a try..:) –  Iyengar Jun 25 '12 at 10:14
    
I can't give you the link unfortunately. The link is not working , may be the server is down. It was present in his website here . But in case if you can't access the article, feel free to drop down your email id and I will be happy in sending you the article. Thank you –  Iyengar Jun 25 '12 at 10:16

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