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angelm Joined: 2006-01-20 Posts: 110	zmaster & mrcun ?? boleh cari?? DIOPHANTINE EQUATIONS The most generally enduring problem of number theory is probably that of diophantine equations. Greek mathematicians were quite adept at solving in integers x and y the equation ux + by = c, wherea, b, and c are any given integers. The close relation with the greatest common divisor algorithm indicated the necessity of treating unique factorization as a primary tool in the solution of diophantine equations. The Greek mathematicians gave some sporadic attention to forms of the more general equation (1) f(x,y)=Ax2+Bzy+Cy2+Dx+Ey+F=O, but achieved no sweeping results. They probably did not know that every equation of this kind cari be solved �completely� by characterizing a11 solutions in a finite number of steps, although they had success with special cases such as x2 -3y2 = 1. In fact, they used continued fraction tech- niques in both linear and quadratic problems, indicating at least esthetically a sense of unity. About 1750 Euler and his contemporaries became aware This section presupposes some familiarity with elementary concepts of group, congruence, Euclidean algorithm, and quadratic reciprocity (which are reviewed in Chapter 1). 1  2 INTRODUCTORY SURVEY of the systematic solvability in a finite number of steps. Yet it was not until 1800 that Gauss gave in his famous Disquisitiones Arithmeticae the solution that still remains a mode1 of perfection. Now a very intimate connection developed between Gauss�s solution and quadratic reciprocity, making unique factorization (in the linear case) and quadratic reciprocity (in the quadratic case) parallel tools. Finally, about 1896, Hilbert achieved the reorganization of the quadratic theory, making full use of this coincidence and thus completing the picture. MOTIVATING PROBLEM IN QUADRATIC FORMS The first step in a general theory of quadratic diophantine equations was probably the famous theorem of Fermat (1640) relating to a (homogeneous) quadraticform in x, y. A prime number p is representable in an essentially unique manner by the form x2 + y2 for integral x and y ifand only ifp = 1 modula 4 (orp = 2). It is easily verified that 2 = l2 + 12, 5 = 22 + 12, 13 = 32 + 22, 17 = 42 + 12, 29 = 5~~+ 22, etc., whereas the primes 3, 7, 11, 19, etc., have no such representation. The proof of Fermat�s theorem is far from simple and is achieved later on as part of a larger result. At the same time, Fermat used an identity from antiquity: (x2 + y2)(2�2 + y�2) = (xx� -yy�) + (xy� + x�y)2, easily verifiable, since both sides equal x2x/2 + Y~�~ + x12y2 + z%�~. He used this formula to build up solutions to the equation (2) x2 + y2 = m for values of M which are not necessarily prime. For example, from the results 3s + 2s = 13, (x = 3, y = 2), 2s + 1s = 5, (5� = 2, y� = l), we obtain 72 + 42 = 65, (xx� -yy� = 4, xy� + x�y = 7). If we interpret the representation for 13 as (-3)s + 2s = 13 (x = -3, y = 2) whereas 2s + 1s = 5, (2� = 2, y� = l), then we obtain (-8)2 + l2 = 65, (xx� -yy� = -8, xy� + x�y = 1);  MOTIVATING PROBLEM IN QUADRATIC FORMS but the reader cari verify that 65 = 72 + 42 = 82 + l2 are the only repre- sentations obtainable for 65 in the form x2 + y2, to within rearrangements of summands or changes of sign. If we allow the trivial additional oper- ation of using (x, y), which are not relatively prime ((k~)~ + @Y)~ = k2m), we cari build up a11 solutions to (2), from those for prime m. Thus Fermat�s result, stated more compactly, is the following: Let Q(x, y> = x2 + Y2. Then a11 relatively prime solutions (x, y) to the problem of representing Qk Y> = m for m any integer are achieved by means of the successive application of two results called genus and composition theorems. GENUS THEOREM (3) Q@, Y) = P cari be solved in integral x, y for p a prime of and only ~fp G 1 (mod 4), or p = 2. The representation is unique, except for obvious changes of sign or rearrangements of x and y. COMPOSITION THEOREM (4) Q(x, Y> QC%�,Y�> = Q@x -YY�, S�Y + ~y�>. In the intervening years until about 1800, Euler, Lagrange, Legendre, and others invented analogous results for a variety of quadratic forms. Gauss (1800) was the first one to see the larger problem and to achieve a complete generalization of the genus and composition theorems. The main result is too involved even to state here, but a slightly more difficult special result Will give the reader an idea of what to expect. (See Chapter XIII.) Let Q,(x, Y> = x2 + 5y2, Q&, Y) = 2x2 + 2s~ + 3y2. Then a11 relaticely prime solutions (x, y) to the problem of representing Q, Posted on 2006-04-16 13:40:31
Author	Comments
angelm Joined: 2006-01-20 Posts: 110	one common trend for 6/42 0r 6/69 type TOTO games is, mean or average of everty 11 drawn numbers, 3 numbers will repeat. Posted on 2006-04-16 19:31:25
monyetekor Joined: 2005-03-20 Posts: 157	asan, org tua zaman dulu pun pakai formula ini... hanya mereka tidak tau nama nyer jer... system2 ni bukan baru... dah wujud dari zaman dulu... skang ni pakai computer... tu beza antara dulu dan kini. tq mE Posted on 2006-04-16 19:28:34
angelm Joined: 2006-01-20 Posts: 110	Magnum kah, Toto kah, Singapore 4D kah, on & on & on Final result are derive from a specific formula being programmed into thier computer system. these formulas can be change anytime or just a random pick numbers to post as results for each draw, each day, each date, each month, each year, each date, each...... TQ Posted on 2006-04-16 19:26:16
asan_kerbo Joined: 2006-03-10 Posts: 471	orang tua zaman dulu tak guna formula camni main.. kena jugak ribu2.. apa2pun, thanks for sharing. Posted on 2006-04-16 19:19:48
budak_kecik Joined: 2005-11-20 Posts: 572	Posted on 2006-04-16 16:37:19
zmaster Joined: 2006-02-12 Posts: 543	thanks monyetekor... erdos conjecture only.... whatever... anyway, i'm not specialist in Mathematic... so, better mr anglem asked whos specialization in math theory... he heh he... zmaster_oracle Posted on 2006-04-16 15:40:26
monyetekor Joined: 2005-03-20 Posts: 157	Erdos conjecture not Erdos Stewart theory mE Posted on 2006-04-16 15:19:41
zmaster Joined: 2006-02-12 Posts: 543	opss... sorry, this is by Erdos Stewart theory, For me, they has so many problem in the elementary theory of numbers and the representations of integers by binary forms..... so i juz solve your problem, not answer your questions... he he he he... zmaster_the oracle... Posted on 2006-04-16 15:11:41
zmaster Joined: 2006-02-12 Posts: 543	to mr anglem, okey, Introduction, u want to showed that, certain diophantine equations have many solutions. In this way they indicated how far certain results are capable for improvements at most. First we mention some relevant results from the literature on upper bounds for the numbers of solutions of Diophantine equations and then we sketch how our method leads to opposite results. 1. Thue and Thue-Mahler equations. Let f(x; y) = a 0 x n + a 1 x n\Gamma1 y + ::: + an y n 2 Z [x ; y ] be a binary form (i.e. homogeneous polynomial) of degree n Posted on 2006-04-16 15:06:08
monyetekor Joined: 2005-03-20 Posts: 157	angelm, A Diophantine equation is an equation in which only integer solutions are allowed. this equation was proven inaccurate by Yuri Matiyasevich in 1970. however, Ogilvy and Anderson (1988) give a number of Diophantine equations with known and unknown solutions. when applied to number theory it has an infinite number of solutions in the positive integers... proven. I suggest you look at kernels too. glad to see someone on the forum has the same vibrations. let's take this to email... if you are agreeable. monyetekor at y..a..h..o..o dot c..o..m Posted on 2006-04-16 14:54:38
seritujuh Joined: 2006-04-09 Posts: 66	nie mato pelajaran apo lak niee....??? Posted on 2006-04-16 14:24:10