Notes on the Fundamental Theorem of Arithmetic
... attendant performs a sequence of operations T1 , T2 , . . . , Tn whereby with the operation Tk , 1 ≤ k ≤ n, the condition of being locked or unlocked is changed for all those lockers and only those lockers whose numbers are multiples of k. Show that after all the n operations have been performed, al ...
... attendant performs a sequence of operations T1 , T2 , . . . , Tn whereby with the operation Tk , 1 ≤ k ≤ n, the condition of being locked or unlocked is changed for all those lockers and only those lockers whose numbers are multiples of k. Show that after all the n operations have been performed, al ...
THE DIVISOR PROBLEM ON SQUARE
... Could the bound be reduced? The answer is definite. Hence we have studied further more the asymptotic properties of and have got some non-trivial properties about it. About this problem, we know very little at present. At least we have not found it in any reference that we could find. Therefore, in ...
... Could the bound be reduced? The answer is definite. Hence we have studied further more the asymptotic properties of and have got some non-trivial properties about it. About this problem, we know very little at present. At least we have not found it in any reference that we could find. Therefore, in ...
Introduction to Algebraic Proof
... To prove that the product of three consecutive positive integers is a multiple of 6 we must choose the general case of three consecutive positive integers: k, k+1 and k+2 and use the properties of division of integers. First we check for the presence of even integers. The integer k can either be eve ...
... To prove that the product of three consecutive positive integers is a multiple of 6 we must choose the general case of three consecutive positive integers: k, k+1 and k+2 and use the properties of division of integers. First we check for the presence of even integers. The integer k can either be eve ...
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[math.NT] 4 Jul 2014 Counting carefree couples
... It is well known that the probability that an integer is squarefree is 6/π 2 . Also the probability that two given integers are coprime is 6/π 2 . (More generally the probability that n positive integers chosen arbitrarily and independently are coprime is well-known [17, 22, 27] to be 1/ζ(n), where ...
... It is well known that the probability that an integer is squarefree is 6/π 2 . Also the probability that two given integers are coprime is 6/π 2 . (More generally the probability that n positive integers chosen arbitrarily and independently are coprime is well-known [17, 22, 27] to be 1/ζ(n), where ...
Solutions to polynomials in two variables
... appears in Fermat’s last theorem, which says that for n ≥ 3, there are no positive integers a, b, and c such that an + bn = cn , or equivalently, there are no rational numbers ac , a n c ...
... appears in Fermat’s last theorem, which says that for n ≥ 3, there are no positive integers a, b, and c such that an + bn = cn , or equivalently, there are no rational numbers ac , a n c ...
Practice Questions
... • If f is one-to-one, f ◦ g is one-to-one. • If f and g are both onto, then f ◦ g is onto. 4. Find gcd(2n + 1, 3n + 2), where n is a positive integer. Hint: Use the Euclidean algorithm. 5. Define a function g on the non-negative integers by g(0) = 2, g(1) = 3 and g(n + 1) = 3g(n) − 2g(n − 1) for al ...
... • If f is one-to-one, f ◦ g is one-to-one. • If f and g are both onto, then f ◦ g is onto. 4. Find gcd(2n + 1, 3n + 2), where n is a positive integer. Hint: Use the Euclidean algorithm. 5. Define a function g on the non-negative integers by g(0) = 2, g(1) = 3 and g(n + 1) = 3g(n) − 2g(n − 1) for al ...
mn is a lattice point if ,mn∈ (2,3)
... Let d be a common divisor of a and b . Then d= a ∧ b if and only if d can be written as a linear combination of them, namely, if and only if ax + by = d has a solution. Sylvester’s Theorem. Let a and b be positive integers with a ∧ b = 1 . Then c = ab − a − b is the largest value of c for which the ...
... Let d be a common divisor of a and b . Then d= a ∧ b if and only if d can be written as a linear combination of them, namely, if and only if ax + by = d has a solution. Sylvester’s Theorem. Let a and b be positive integers with a ∧ b = 1 . Then c = ab − a − b is the largest value of c for which the ...
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"".