We reduced the unsolved problem of Evans triangles to the problem of nding rational points on a class of elliptic curves, the twisted Edwards curves En. We proved that the set of all Evans triangles with a given ratio is isomorphic to a quotient group of En(Q). After this, using Sage, we can solve the Evans problem for an amount of cases within the capacity of computer. Even without the help of a computer, we can still obtain a large class of new Evans triangles, and nd an eective way to test if the rank of an Edwards curve is positive or not.

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In this paper, we will discuss some upper bound formulas for Ramsey numbers.At first, we will derive a more unified form from two parameter formulas which was obtained by Yiru Huang etc. Then, similar to their methods, we introduce two new bound formulas for Ramsey numbers in this paper. At last, as the applications of our new bound formulas, some upper bounds for small Ramsey numbers will be obtained.

Quotient sets have attracted the attention of mathe- maticians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gauss- ian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadratic ring is dense in the complex plane and the set of quotients of primes in a real quadratic number ring is dense in R: An interesting open question is introduced by Sittinger: Is the set of quotients of Hurwitz primes dense in the quaternions? In this paper, we answer the question and prove that the set of all quotients of Hurwitz primes is dense in the quaternions.