Nicolae Ion Bratu



Department of Mathematics - Craiova University - Romania

Address: Th. Aman Str., bl.Casa Alba - 1100 Craiova - Romania

Phone: 0040 251 415103

E-mail: bratu@oltenia.ro

  • Research Interests:

  • Interests:

  • Papers and research:

    • [1] On the Quaternary Quadratic Diophantine Equations (2)(pdf) - publish with Ben Bratu in "Bulletin of Pure & Applied Sciences" - Vol.-19-E No.2-2000

    [2] A Generalization of Gauss Theorem on Quadratic Forms(pdf) - publish with Adina Cretan in "Bulletin of Pure & Applied Sciences" - Vol.-21-E No.1-2002

    [3] Eseu asupra Ecuatiilor Diofantice - Ed. Adel Craiova 1994

    [4] Note de analiza diofantica - Ed. Dutescu Craiova 1996

    [5] On the Quaternary Quadratic Diophantine Equations (1) - Conf. on Smarandache F. Notions. American Res. Press-1997

    [6] O Afirmatie mai Tare pentru Criteriul lui Grunert - Gazeta Matematica, Nr.3-4, 1991, Bucuresti

    [7] “On the cubic combination and the third degree Ramanujan identities”(pdf) - publish with Adina Cretan in “Varahmihir Journal of Math.- Sc.-Canada - Vol.6-2006 - Nr. 1

    [8] Disquisitiones Diophanticae(pdf) - Ed. Reprograph - Craiova - 2006

    RESULTS:

    A.  Lagrange's Four-Square Theorem
    The number of 4 squares is reduced only to 3 distinct squares, without any exception (in the Legendre's Theory there is an exception)
    The Theorem of the Three Distinct Squares - Bratu
    Every number is the sum of three squares, or of three squares with one duplicated. Further, numbers of the form
    22k (8 l + 7) are only of the second type, numbers of the form 22k+1 (81 + 7) are only of the first type, while numbers of neither of these two forms are of both types.
    For any natural number z, there are at least three integer numbers (u, v, w) or/and (a,b,c), in order to have representations:

    z = u2 + v2 + w2 (a)
    z = a2 + b2 + 2c2 (b)

    For z = z1 = 22k (81 + 7),   we have only the representation (b),
    for z = z2 = 22k+1 (81 + 7),   we have only the representation (a), and ,
    for z ¹ z1 , z¹ z2,   we have, in the same time, the representations (a) and (b).
    Examples:
    z1 = 15,   we have z1 = 32 + 22 + 2 * 12 (b)
    z2 = 30,   we have z2 = 52 + 22 + 12 (a)
    z3 = 21,   we have z3 = 42 + 22 + 12 (a) and   z3 = 32 + 22 + 2 * 22 (b).
    The proof results using the function "quadratic combination", Lemma 2 and noticing the graph of the equation
    E32 : x2 + y2 + t2 = z2
    See papers: [1],...[5].

    B.  Gauss's Theorem for Binary Quadratic Forms
    Bratu proves a general theorem of the Gauss's type, for any quadratic forms.
    The proposed method, for the determination of the solutions of quadratic equations, is different from the ones that exist in literature.
    See paper [2].

    C.  On Quaternary Quadratic Diophantine Equation of type Euler - Carmichael- Mordell:  x2 + b*y2 + c*z2=w2  (CM), where b,c are integers
    Bratu finds the general solution with four parameters of the equation (CM).
    See [1],...[5].

    D.  Function "Quadratic Combination"
    Definition: Quadratic combination is a numerical function Q, which associates to the both solutions from G22 four solutions from G32;
    symbolically we have:   Q : G22 x G22 ® G32.
    We noted G22 the complete system of solutions for the equation: x2 + y2 = z2 and, also, G32 for the equation: x2 + y2 + z2 = w2.

    See paper [3].
    E.  The Bratu's Identities and the Bratu's Lemma
    Starting from Bratu's Identity, I proved the following Bratu's Lemma:
    Given two solutions of the complete system of solutions of the homogenous ternary equation (3), out of the two solutions there can be generated four solutions - that can be also equal two by two - for each of the four quaternary solutions (24). It named the four equations "twin equations".
    It is proved that the Theorem of the Three Distinct Squares resulting naturally from the function Quadratic Combination, a function that, it its turn, is a consequence of the Bratu's Identity and of the Bratu's Lemma.

    F. About the Last Theorem of Fermat
    In the part II of the work [8], we shall present succinctly and schematized the content of the "Memorandum to the Romanian Academy" concerning the Great Fermat Theorem. But we will publish, for the first time, a new lemma representing a completion of the arithmetic method proposed by us to prove the Last Fermat Theorem. If the method called "g.r.s." allowed us to pass from the cyclotomic to the quadratic field, through the Lemma demonstrated now, one can pass to the rational field, where the fundamental theorem of the arithmetic has validity.
    It is proposed and proved the following Lemma Euler-Legendre-Bratu:

    G. In the paper [7] we have found three results
    a) A new function, called "Cubic Combination";
    b) A set of new Romanujan type identities;
    c) The Miller-Woulett conjecture (1955 - proposed) was proved

    H. The graphic representation of solutions of the quadratic equations
    In [3](1994) , than in [4], [2], [5] and [8], we demonstrated a general Lemma and the equivalent form of Lemma is the following:
    Lemma 1b- Set F² of the E² equation solutions is isomorphic with the set of nodes of the G² oriented graph, defined by a recurrence relation
    Examples:
    1) The Pell equation- In the graphic representation, the set of natural solutions is a chain;
    2) The Pythagorean equation - The set of primitive solutions is a tree;
    3) The homogenous quadratic quaternary equation - The primitive solutions are representation through the nodes of a oriented graph;