Wilson`s Theorem and Fermat`s Theorem
... (b) Since (m, n) = 1, I may find x such that mx = 1 (mod n). Since (ai , n) = 1, so I may find bi such that ai bi = 1 (mod n). Then (xbi )(ami ) = (mx)(ai bi ) = 1 (mod n), which proves that ami is invertible mod n. Hence, (ami , n) = 1 — the ma’s are relatively prime to n. Now if mai = maj (mod n), ...
... (b) Since (m, n) = 1, I may find x such that mx = 1 (mod n). Since (ai , n) = 1, so I may find bi such that ai bi = 1 (mod n). Then (xbi )(ami ) = (mx)(ai bi ) = 1 (mod n), which proves that ami is invertible mod n. Hence, (ami , n) = 1 — the ma’s are relatively prime to n. Now if mai = maj (mod n), ...
MATH 25 CLASS 16 NOTES, OCT 26 2011 Contents 1. Fast
... one, but also is one that is beyond the scope of this class. Another interesting question is how many Carmichael numbers there are. As a matter of fact, it is a difficult theorem of Alford, Granville, and Pomerance in 1992 that there are infinitely many Carmichael numbers, and they show that there a ...
... one, but also is one that is beyond the scope of this class. Another interesting question is how many Carmichael numbers there are. As a matter of fact, it is a difficult theorem of Alford, Granville, and Pomerance in 1992 that there are infinitely many Carmichael numbers, and they show that there a ...
Fermat's Last Theorem
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 can satisfy the equation an + bn = cn for any integer value of n greater than two. The cases n = 1 and n = 2 were known to have infinitely many solutions. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The theretofore unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics and prior to its proof it was in the Guinness Book of World Records for ""most difficult mathematical problems"".