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I saw a paper by Manjul Bhargava that he solved the Conway-Schneeberger fifteen theorem by lattices . Do we have any other solution for this theorem from different way ? And how can we generalize this theorem for other forms ? Thanks

Reference : " on the Conway-Schneeberger fifteen theorem " paper by Bhargava

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    You should add a reference to the paper by Bhargava you are citing. (You should also have added the (reference-request) tag to your question, but I have done that for you.) – 2017-02-25
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    Have you looked at https://en.wikipedia.org/wiki/15_and_290_theorems and the references therein? – 2017-02-26

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I don't know of any other ways to prove the result, but I wouldn't expect one since a statement like "if it represents the first $X$ numbers, then it represents all the rest" is very analytic in flavor. There is a lot of work in quadratic forms, and algebraic methods typically give conditions of a different nature**

The theorem has been generalized! In fact, it was extended to all positive definite integral quadratic forms (including those that can have half-integers in their matrix representation) in the Hanke-Bhargava 290 Theorem, which states that if a quadratic form represents all numbers up to $290$, then it is universal.

There have been a lot of other results asking "when can we guarantee that a positive definite quadratic form represent $S$?"

where $S$ is some infinite subset of $\mathbb N$.

This is a game "we can play" since Bhargava has shown (which may be folklore by this point) that every infinite subset $S$ has a corresponding finite subset that one would need to check to ensure that all of $S$ is represented. Note that all related theorems give sufficient, but also necessary lists of numbers that need to be represented. So, the "smaller" the infinite subset (say, with respect to density) the more difficult the question since there are more quadratic forms that represent most of $S$ than there are representing all of $\mathbb N$.

Here are some similar results:

when a quadratic form represents all odd numbers.

when a quadratic form can except at most two values

when a quadratic form represents all numbers coprime to 3.

The lattices are typically just the language used to "generate" a full list of quadratic forms that may be universal. It plays an important role in deciding where to start with quadratic forms. The main methods for proof of representation in the above results are related to generating functions and modular forms, you can see my answer here for a really coarses approximation of what's involved in some of the papers.

**Here is a notable exception, which is incidentally (to my knowledge) the first professional paper of Paul Halmos! It contains a few errors, but it is genuinely astounding that he obtained the result here.