# Nicolae Ion Bratu

*Department of Mathematics - Craiova University - Romania*

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

Phone: 0040 251 415103

**E-mail:**
*
bratu@oltenia.ro *
*Research Interests:*
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- Numbers Theory
- Diophantine Problems
- Numerical Functions
- Differential Equations
- Cosmology
- Astronomy

* Interests:*
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- Chess
- Belles-lettres
- Engineering Tehnology
- Astrology

*Papers and research:*
#### [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

**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

2^{2k} (8 l + 7) are only of the second type, numbers of the form 2^{2k+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 = u^{2} + v^{2} + w^{2} (a)

z = a^{2} + b^{2} + 2c^{2} (b)

For z = z_{1} = 2^{2k} (81 + 7), we have only the representation (b),

for z = z_{2} = 2^{2k+1} (81 + 7), we have only the representation (a), and ,

for z ¹ z_{1
}, z¹ z_{2}, we have, in the same time, the representations (a) and (b).

Examples: z_{1} = 15, we have z_{1} = 3^{2} + 2^{2} + 2 * 1^{2} (b)

z_{2} = 30, we have z_{2} = 5^{2} + 2^{2} + 1^{2} (a)

z_{3} = 21, we have z_{3} = 4^{2} + 2^{2} + 1^{2} (a) and
z_{3} = 3^{2} + 2^{2} + 2 * 2^{2} (b).
The proof results using the function "quadratic combination", Lemma 2 and noticing the graph of the equation

E_{3}^{2} : x^{2} + y^{2} + t^{2} = z^{2}

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: * x^{2} + b*y^{2} + c*z^{2}=w^{2} (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 G_{2}^{2} four solutions from G_{3}^{2};

symbolically we have:
*Q* : G_{2}^{2} x G_{2}^{2} ® G_{3}^{2}.

We noted G_{2}^{2} the complete system of solutions for the equation: x^{2} + y^{2} = z^{2} and, also, G_{3}^{2} for the equation: x^{2} + y^{2} + z^{2} = w^{2}.

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;*