Nicolae Ion Bratu (act.2017)


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 and

.Profile (in Romanian)


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


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


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 .

Research Interests:


.Papers and research:

AA= Numbers Theory

1. BRATU I..N.- Disquisitiones Diophanticae- Ed. Reprograph- 2006

2. BRATU I..N.- On the Quaternary Quadratic Diophantine Equations (1) - Conf. on Smarandache F. Notions. American Res. Press-1997; p.147-150

3 BRATU I.N. and BRATU N.B and CRETAN N.A. - On the Quaternary Quadratic Diophantine Equations (2a) "Bulletin of Pure & Applied Sciences" - Vol.-19-E -2000

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

6 BRATU I..N.-"Eseu asupra Ecuatiilor Diofantice -Essay on Diofantine Equations” Ed. Adel Craiova 1994

7. BRATU I..N- “Note de analiza diofantica- Diofantic analysis notes”;- Ed. Dutescu Craiova 1996

8. BRATU I..N-"The Theorem of the Three Distinct Squares-(3)" - pg. 628-641 - "Octogon Math. Magazine"-Vol.15-2007- Romania

BRATU I. N. and CRETAN N. A: A Generalization of Gauss Theorem on Quadratic Forms(pdf”) - "Bulletin of Pure & Applied Sciences" - Vol.-21-E No.1-2002

10. BRATU I..NO Afirmatie mai Tare pentru Criteriul lui Grunert” - Gazeta Matematica, Nr.3-4, 1991, Bucuresti

11. BRATU I..N "The Theorem of the Three Distinct Squares-3" - pg. 628-641 - "Octogon Math. Magazine"-Vol.15-2007- Romania

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 !

12. BRATU I .N. and CRETAN N. A: “ On the cubic combination and the third degree Ramanujan identities””(pdf) - publish in “Varahmihir Journal of Math.- Sc.-Canada - Vol.6-2006 - Nr. 1

13 BRATU I .N.:”O Afirmatie mai Tare pentru Criteriul lui Grunert- A stronger assertion for Gruunnert's criterion” - Gazeta Matematica, Nr.3-4, 1991, Bucuresti

14. BRATU I .N.: "Memoriu - Probleme nerezolvate in teoria numerelor--

Memorandum unresolved problems in number theory”- Academia Romana; Acad. N.T - 1983 - Nepublicat – Romania;

NOTE 2: Lucrarea avea 3 capitole: 1. Marea Teorema Fermat; 2. Teorema Lagrange a celor patru patrate; 3. Alte probleme nerezolvate

15 BRATU I .N.: "On the Fermat`s Last Theorem. A new Proof for the Cases n=3 and n=5" - pg. 934-945 - "Octogon Math. Magazine"-Vol.16- 2008- Romania;

NOTE 3: A= Resuming the proof Last Theorem for the cases n- 3 and n=5 –works 5-6

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

BB= Other research

[17] BRATU I..N.- “Legile lui Kepler. The formation of planetr”- Ed. Adel Craiova 1995

[18] BRATU I..N.- “Black holes ?There can be no black holes”- Ed. Adel Craiova 1995

[19] BRATU I..N.-"Program de calcul seismic constructii Caplan- Algeria I.P.J.D - 1981 - Romania

[20] BRATU I..N. and MUNTEANU G. E. -"Noua metoda de dimensionare compresune excentrica oblica Revista Constructii - 1984 - Romania

[21] "Proiect examen de stat: Amplidina- Aspecte noi si optimizarea proiectarii" " Academia Romana- Acad.A.M.- martie 1961


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) 
” E
very 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 = a
2 + 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 z1 and z z2, have, in the same time, the representations (a) and (b). 

z1 = 15,   we have z1 = 32 + 22 + 2 * 12 (b)
2 = 30,   we have z2 = 52 + 22 + 12 (a)
3 = 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 
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:  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],...[6]. 

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 [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 F of the E equation solutions is isomorphic with the set of nodes of the G oriented graph, defined by a recurrence relation

1) The Pell equationIn 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;