The Diophantine X~2+13=y~n

The diophantine equation in number theory is one of the oldest branches of history many famous mathematical problems related to such equations. Since ancient times, many mathematicians have made outstanding contributions to this. It is these significant results not only enriched the content of indefinite equation itself, but also modern discrete mathematics, algebraic geometry, combinatorics, cryptography has laid a foundation. In particular the introduction of the ideal mathematician kummer number of concepts, making the research related to this breakthrough has been made.Promotion of the indeterminate equation, algebraic number theory has been the initial formation and development. In the second domain, the ideal, unit number, type and number of the indeterminate equation plays an important role in the research. The equation x~2 + D = y~n, in the second domain, if the number of classes Q ( (?) )is 1, the only ring of integers can be used directly to solve the decomposition theorem; in the second domain, if not the number of classes is not 1, the class discussed in this paper a number of classes is 2, the first use of the ideal number of unique decomposition theorem classified discussion, re-use a power of induction, with the remaining properties and the nature of quadratic fields obtained equation.In this paper, algebraic number theory, the main research of the indefinite equation x~2 + 13 = y~n( 0 < y< 100) Solution of the case, VII contains four chapters. The first Chapter Overviews the diophantine equation x~2 + D = y~n research status and main contents of this article. The second chapter describes some of the main definitions, properties, theorems. The third chapter was proved during the diophantine equation x~2 + 13 = y~n( 0 < y< 100), which contains three sections. Section I proved that the diophantine equation x~2 + 13 = y~n( 0 < y< 100)has solutions of the necessary and sufficient conditions. Section II proved that if is an odd number, that is y = 17,29,49,77, the diophantine equation x~2 + 13 = y~n( 0 < y< 100)was the situation. Section III show that if even, that is y = 14,22,38,62,94, the diophantine equation x~2 + 13 = y~n(0 < y < 100) was the situation.