Nicolae Ion Bratu
Department of Mathematics - Craiova University - Romania
Address: Th. Aman Str., bl.Casa Alba - 1100 Craiova - Romania
Phone: 0040 251 415103
- Numbers Theory
- Diophantine Problems
- Numerical Functions
- Differential Equations
Papers and research:
- Engineering Tehnology
 Eseu asupra Ecuatiilor Diofantice - Ed. Adel Craiova 1994
 Note de analiza diofantica - Ed. Dutescu Craiova 1996
 On the Quaternary Quadratic Diophantine Equations (1) - Conf. on Smarandache F. Notions. American Res. Press-1997
 O Afirmatie mai Tare pentru Criteriul lui Grunert - Gazeta Matematica, Nr.3-4, 1991, Bucuresti
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: ,....
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 .
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).
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 .
E.  The Bratu's Identities and the Bratu's Lemma
from Bratu's Identity, I proved the following Bratu's
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
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 , 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
G. In the paper  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
In (1994) , than in , ,  and , 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
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
3) The homogenous quadratic quaternary equation - The primitive solutions
are representation through the nodes of a oriented graph;