Some remarks on Euler`s totient function - HAL
... Clearly α0 . . . αr is k written in binary form. Therefore there can be at most one odd number n such that φ(n) = 2k . If αi = 1 only when Fi is prime, then there exists n odd such that φ(n) = 2k and O(2k ) = 1. On the other hand, if there is an αi such that Fi is not prime, then there does not exis ...
... Clearly α0 . . . αr is k written in binary form. Therefore there can be at most one odd number n such that φ(n) = 2k . If αi = 1 only when Fi is prime, then there exists n odd such that φ(n) = 2k and O(2k ) = 1. On the other hand, if there is an αi such that Fi is not prime, then there does not exis ...
ON CONGRUENCE PROPERTIES OF CONSECUTIVE VALUES OF
... number of parts t such that 2 ≤ t < ". We state this result in the following theorem. Theorem 3 For " an odd prime, m ≥ 0, and 2 ≤ t ≤ ", p(m · lcm("), t) ≡ 0 (mod "). 1.1 Examples of Theorems 2 and 3, and Corollary 2. What is surprising about Theorem 2 and Corollary 2 is that they establish congrue ...
... number of parts t such that 2 ≤ t < ". We state this result in the following theorem. Theorem 3 For " an odd prime, m ≥ 0, and 2 ≤ t ≤ ", p(m · lcm("), t) ≡ 0 (mod "). 1.1 Examples of Theorems 2 and 3, and Corollary 2. What is surprising about Theorem 2 and Corollary 2 is that they establish congrue ...
Waring`s problem, taxicab numbers, and other sums of powers
... 1. Introduction. Many of the most perplexing problems in number theory arise from the interplay of addition and multiplication. One important class of such problems is those in which we ask which numbers can be expressed as sums of some numbers which are defined multiplicatively. Such classes of numb ...
... 1. Introduction. Many of the most perplexing problems in number theory arise from the interplay of addition and multiplication. One important class of such problems is those in which we ask which numbers can be expressed as sums of some numbers which are defined multiplicatively. Such classes of numb ...
135. Some results on 4-cycle packings, Ars Combin. 93, 2009, 15-23.
... The study of primes plays the most important role in Number Theory. It is well-known that the number of primes is infinite and also for each positive integer n there are n consecutive integers which are not primes. Therefore, we may have a very long sequence of integers which contains no primes. But ...
... The study of primes plays the most important role in Number Theory. It is well-known that the number of primes is infinite and also for each positive integer n there are n consecutive integers which are not primes. Therefore, we may have a very long sequence of integers which contains no primes. But ...
Pythagorean Triples. - Doug Jones`s Mathematics Homepage
... We now know that one side of a Pythagorean triple right triangle must be odd. So give me any odd number, and I can fairly quickly give you back a Pythagorean triple with that odd number as a side.11 Here’s how it works. Let n be any odd number (odd positive integer). Then square it, subtract one, an ...
... We now know that one side of a Pythagorean triple right triangle must be odd. So give me any odd number, and I can fairly quickly give you back a Pythagorean triple with that odd number as a side.11 Here’s how it works. Let n be any odd number (odd positive integer). Then square it, subtract one, an ...
Differential Equations A differential equation is an
... A population of unicorns increases proportionally to its size. On the second day of the experiment, there were 100 unicorns, and on the fourth day, there were 300 unicorns. Approximately how many unicorns were in the original population? Let P = P (t) be the number of unicorns at time t. The differen ...
... A population of unicorns increases proportionally to its size. On the second day of the experiment, there were 100 unicorns, and on the fourth day, there were 300 unicorns. Approximately how many unicorns were in the original population? Let P = P (t) be the number of unicorns at time t. The differen ...
I1 Pythagoras` Theorem and Introduction Trigonometric Ratios
... trigonometric tables took place around 500 AD, through the work of Hindu mathematicians. In fact, tables of sines for angles up to 90° were given for 24 equal intervals of 3 43 ° each. The value of 10 was used for π at that time. Further work a century later, particularly by the Indian mathematician ...
... trigonometric tables took place around 500 AD, through the work of Hindu mathematicians. In fact, tables of sines for angles up to 90° were given for 24 equal intervals of 3 43 ° each. The value of 10 was used for π at that time. Further work a century later, particularly by the Indian mathematician ...
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"".