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I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless.

My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.

I came with some ideas like the Turing machine but I think I'm not grasping the right examples.

Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'?

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Utility is defined vis-a-vis context. The complex numbers were invented to solve $x^2+1=0$, which is useless in any application restricted to the reals (but turn out to be useful in many other ways unforeseen at the time of their construction). –  Arkamis Dec 17 '12 at 18:24
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Binary numbers have a long history, but were not very "useful" until scientists began applying them to electronic circuits and computers. en.wikipedia.org/wiki/Binary_numeral_system –  Andrew Dec 17 '12 at 18:28
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Ramsey theory isn't particularly useful in real world applications (although it can be useful in proofs for other branches of mathematics). However, I often joke that often in data analysis problems, one practices "applied Ramsey theory" -- finding the one data set out of many that perfectly matches one's algorithm. –  Arkamis Dec 17 '12 at 18:42
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I once read or heard that Sophus Lie claimed that his "Lie groups" were totally useless. It's one of the cornerstones of modern particle physics now. –  Raskolnikov Dec 17 '12 at 18:45
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@EdGorcenski I believe that historically, the complex numbers were invented to solve cubic equations, not quadratic ones. –  MJD Dec 17 '12 at 21:43
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The Quaternions were considered useless for a long time.

Anyhow, the set of all unit quaternions is a double cover of $SO_3(\mathbb R)$. This allows us to represent any rotation matrix by a quaternion, which is used now in computer games (instead of using 9 parameters to parametrize a rotation matrix, we can use only 3 for the quaternions).

You can read more here: http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation

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I'm not so sure if Quaternions actually qualify. There was sort of a religion of "quaternionists" led by Macfarlane. See the Quaternion Society and frowned upon publicly by Klein, see here. The fact that they yielded a neat formulation of Hamilton's equations and a natural proof of the four squares identity was observed shortly after their conception, if I'm not completely mistaken. –  Martin Dec 19 '12 at 21:51
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Quote from G. H. Hardy1

The Theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics.

Just 30 years after his death, the RSA algorithm was introduced which is deeply rooted in number theory and is now important part of sending encrypted information electronically, e.g., over the Internet.

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