Αποτελέσματα Αναζήτησης
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2.
ON FERMAT’S LAST THEOREM FOR N = 3 AND N = 4 R. ANDREW OHANA Abstract. A solution to Fermat’s equation, xn + yn = zn, is called trivial if xyz = 0. In this paper we will prove Fermat’s Last Theorem, which states all rational solutions are trivial for n > 2, when 3 jn or 4 jn. For n = 3 we will show all solutions in the Eisenstein Field, Q(p
Fermat's last theorem (also known as Fermat's conjecture, or Wiles' theorem) states that no three positive integers x,y,z x,y,z satisfy x^n + y^n = z^n xn + yn = zn for any integer n>2 n> 2.
Gauss showed that a regular n-gon is con-tructible with a compass and a straightedge if and only if n is a power of 2 times a product of distinct Fermat primes. Here are some properties of the Fermat numbers. Proposition. If p is prime and p | Fn, then p = k · 2n+2 + 1 for some k.
A Fermat number Fn = 2 6 Ù+ 1 (for n ≥ 1) can be thought of as a square whose side length is 2 6 Ù 7 - plus a unit square (see figure1). Hence, determining whether a (Fermat) number is a composite or not is equivalent to determining whether we can rearrange the unit-square blocks to form a rectangle (see figure2).
This gives a solution to the Fermat equation with exponent p. If n has no odd prime factors then n is a power of 2; since n ≥ 3 we must have n = m·4, so (xm)4 +(ym)4 = (zm)4. Therefore it suffices to show that the Fermat equation has no nonzero solutions for prime exponents and for the exponent 4.
equation reads X4 + Y4 = Z2; which is just the equation we started from. So from any solution (x;y;z) of the equation x4 + y4 = z2 with gcd(x;y;z) = 1, x;y > 0 and x even, we obtain a second solution (X;Y;Z) with gcd(X;Y;Z) = 1, X;Y > 0 and X even, where x = 2X2Y; y = X4(1 4Y4); z = X4(1 + 4Y4):
