Incerc sa propun o problema intr-o tematica cu meritul ca va mai trai o vreme buna. Asa cum stau lucrurile nu pot decat propune o problema de rezolvat cu calculatorul, dar poate ca exista observatii in cazuri speciale care aduc explicit solutia.

Iata despre ce este vorba:
Pe pagina
http://en.wikipedia.org/wiki/Congruent_number
se defineste notiunea de numar congruent. Definitia simpla dintr-un domeniu "elementar" poate induce in eroare asupra complexitatii problemei...

Definitie:
Un numar rational pozitiv n se numeste numar congruent
daca si numai daca
exista un triunghi dreptunghic cu laturile a,b (catete) si c (ipotenuza) numere rationale, triunghi care are aria n.

Explicit, dat fiind n avem de rezolvat in numere rationale a,b,c:

aa + bb = cc si
ab/2 = n .

Sa se arate ca daca avem o solutie ca mai sus si scriem
x = n(a+c)/b
y = 2n²(a+c)/b²
atunci are loc relatia:

yy = x(x-n)(x+n) .

Dau un exemplu:
Triunghiul notoriu cu laturile 3,4,5 are aria 6.
Deci 6 este un numar congruent.
Atunci x = 6(3+5) / 4 = 12 si y = 2 6²(3+5) / 4² = 36 si incercam sa vedem daca

36² = (12-6) . 12 . (12+6)

Da, deoarece 6.6.6.6 = 6 . 2.6 . 3.6 .


De unde vine denumirea?
Este bine sa observam ca numerele
(5/2)² - 6 , (5/2)² , (5/2)² + 6 ,
sunt patrate perfecte in progresie aritmetica cu ratia 6.
S-a ales cuvantul "congruent" pentru a face aluzie la o astfel de congruenta.

Sa se arate ca daca numerele rationale x,y cu y nenul satisfac relatia de mai sus, atunci numerele a,b,c definite de

a = (x² - n²) / y ,
b = 2nx / y ,
c = (x² + n²) / y .

produc un triunghi dreptunghic de arie n, deci n este congruent.

Sa se arate ca s este un numar rational oarecare,
atunci
n este un numar congruent daca si numai daca
n ss (produsul dintre n si patratul lui s) este un numar congruent.

In acest mod, ne putem reduce usor in studiul numerelor congruente la acele numere rationale n care sunt:
- naturale si
- libere de patrate perfecte.

De aceea si exercitiul "simplu" poate pentru calculator pe care il propun:

Care numere dintre numerele naturale libere de patrate perfecte de la unu la o suta sunt congruente si care nu?

Numerele libere de patrate perfecte sub 100 sunt anume:

(20:04) gp > for( n=1, 100, if( issquarefree(n), print1( n, " " ) ) )
1 2 3 5 6 7 10 11 13 14 15 17 19 21 22 23 26 29 30 31 33 34
35 37 38 39 41 42 43 46 47 51 53 55 57 58 59 61 62 65 66 67
69 70 71 73 74 77 78 79 82 83 85 86 87 89 91 93 94 95 97

Cateva jaloane sunt urmatoarele:
http://en.wikipedia.org/wiki/Congruent_number#Current_progress si
http://en.wikipedia.org/wiki/Tunnell's_theorem

Daca acest ultim "exercitiu" nu isi gaseste publicul, cele de mai sus merita totusi atentie.


Inserez aici cateva documente care poate dezvolta gustul pentru o directie sau alta...
http://modular.math.washington.edu/simuw06/notes/notes.pdf
http://www.williamstein.org/Tables/elliptic_curves_in_nature/congruent_number_problem/congruent_numbers.ps
http://www.rtbot.net/congruent_number
http://www.williamstein.org/Tables/elliptic_curves_in_nature/congruent_number_problem/157.html
http://www.ias.ac.in/resonance/December2009/p1183-1205.pdf
si multe altele.

Citez din ultimul articol...
... Retaining only the squarefree parts of the numbers produced by this procedure,
the first few congruent numbers which show up are
5 ; 6 ; 7 ; 13 ; 14 ; 15 ; 21 ; 22 ; 23 ; 29 ; 30 ; 31 ; 34 ; 37 ; 38 ; 39 ; 41 ; ...
Note that we have not proved that the numbers 1 ; 2 ; 3 are not congruent.
it may simply be that they haven't yet shown up on the list!
Indeed, Leonardo of Pisa (called Fibonacci) (1175-1240)
was challenged to find a rational right triangle of area 5 (he succeeded),
and he conjectured that 1 is not congruent.
This was settled much later by Pierre Fermat (1601-1665).
How can we determine if a specific number such as 157 is congruent?
The naive approach, suggested by the discussion just after Definition 1,
would be to go through a `list' of squares d of rational numbers,
and to see if both d - 157 and d + 157 are squares.
There is indeed such a `list':
first we go through the squares of the finitely many rational numbers
whose numerator and denominator have just one digit,
then through the squares of those (again finitely many)
whose numerator and denominator have at most two digits, and so on.
It turns out that the first square which works for 157,
according to Don Zagier, is

Clearly, this number could not have been found by the naive approach.
Some theory is needed. Also, as before, this approach cannot prove that the given
number, for example 1, is not congruent....