NOTE
Many readers of my works have asked me to publish an explanatory
my profile and add other papers and results, what I will do shortly
now in English and partly in Romanian.
.Department of Mathematics - Craiova University - Romania - RETIRED
.Adress Th. Aman Str., bl.Casa Alba - 1100 Craiova – Romania
.Phone 0040 251 415103 and 040 723 013004
.E-mail bratu@oltenia.ro and mathnib@yahoo.com
.Profile (in Romanian)
STUDII
1946-1953– Scoala elementara- magna cum laude
1951- eliminat din motive politice clasa a VI-a
1956- Liceul Clasic Craiova- magna cum laude
1955- Loc I Olimpiada de Matematica- judet Dolj
1956- Loc I Concurs National Limbi Clasice- Latina si Elina
1961- Facultatea de Electrotehnica Bucuresti
1961- amanat Examenul de Stat din motive politice
1972- Facultatea de Educatie Fizica si Sport Bucuresti
1974- Facultatea de Matematica din Craiova- Media 10
1975- Masterat Calcul Electronic Software- Media 10
1981- Facultatea de Drept Sibiu an I
1982- eliminat din motive politice sem 2- an II
1982- Masterat Calcul Seismic- Rezstenta si Stabilitate- media 10
1995- Doctor Inginer- U.T.C. Bucuresti
2000- Curs EuroCor Astrologie- media Foarte Bine
LUCRARI INGINERESTI
1961-1999- Peste 400 programe de calcul de diverse specialitai, Topo,Retele etc
1982- Autor program calcul structuri- CAPLAN ALGERIAi
1984- Autor metoda de cakcul Metoda de calcul compresiune excentrica oblica
1985- Program de calcul “Retele inelare- Retin”
1986 Program Retele Gaze naturale- exemplu realizat- uras Filiasi
1964- 1998 Peste 1000 de proiecte –LEA MT si JT- CN,pres M,R J ,etc
SAH
NOTE When (2016) I needed a moral help from the Chess sector, to convince the
Academy to publish my work on Number Theory, it suddenly came to my mind that
it was intended that I would be ranked 174 in the trance; ulterior a fost retrasa stirea
I really dropped chess in 1995 - when I was 21st:
1962 – Campion National Sah- neclasificati
1969-1995- Campion absolut Sah Regiunea Oltenia –
din 560 de partied nu am pierdut nici una
1978- 1995-Castigator a 6 turnee intrnationale din cele 8 la care am participat
(a fost interzisa participarea la turnee in Occident din motive politice)
1995-Am abandonat sahul individual in anul 1996 , la decesul mamei mele;
Eram pe locul 21 national dupa coeficientul Elo- 2315 si singurul MF din judetul Dolj .
.Papers and research:
4 BRATU I. N. and CRETAN N. A.-On the quaternary quadratic diophantine equations (2) , University of New SouthWalles- Mathematical Gazette- 2003
5. BRATU I..N.-"Graphs in the Theory of the Quadratic Forms" - pg. 092-103 "Octogon Math. Magazine"-Vol.16-2008- Romania
7. BRATU I..N- “Note de analiza diofantica- Diofantic analysis notes”;- Ed. Dutescu Craiova 1996
NOTE 1: On my works are known 2 cases of plagiarism – a.2006 - a USA citizen
initially recognized the priority of my work, then gave up;
and - worse - 2016- two citizens of the Romanian Academy have insidiously declared
that they have known "long" the Bratu-Lagrange theorems – VERY SAD !
B= No negative reaction; only laudatory appreciations from great mathematicians
16. BRATU I .N.: "The Fermat`s Last Theorem" - pg. 046-059 - "Octogon Math. Magazine"-Vol.19-2011- Romania;
NOTE 4: A=:It is proposed and proved the following Lemma
Bratu– Euler-Legendre ; Through lemma B-E-L and through infinite progeny we get
the Last Theorem of Fermat is true. If the method of generating rational solutions - gs.s.r.-
allowed the transition from the cyclotomic to the pathetic body through Lema Bratu – Euler - Legendre - further proved – himself can transfer the whole issue to the body of numbers
B= I was told "Not me" and the other silence
C= I know what is not clear and I could easily explain it at a Conference
A. The Theorem of the Three Distinct Squares - Bratu
Te new proof for 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)
”
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 ≠
z_{1
}and_{
}z
≠z_{2},
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}
Is
the last word in this issue 1983-1994 –1997
See
papers: [1],...[7]
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 [7] and [8}
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],...[6].
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 [6] [7], [15], [16]
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.
Essentially
this new theorems Bratu summarize:
"For
any integer representation by the sum of squares are sufficient three
integers"
F.
About the Last Theorem of Fermat
In
the part II of the work [1], 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 Lemma
Bratu- Euler-Legendre
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 Bratu-
Euler-Legendre
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 Bratu- Euler-Legendre (1983-1996)-Our contribution to the proof of Fermat Last Theorem
See note 3 and 4
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; Ramanujan found them
intuitively, and we through proof
c)
The Miller-Woulett conjecture (1955 - proposed) was proved
H.
The graphic representation of solutions of the quadratic equations
In
[6](1994) , than in [4], [5], [6] and [1], 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;