Solved and unsolved problems in elementary number theory
... Did Pythagoras invent arithmetic dynamics? Consider the map s : N ∪ {0} → N ∪ {0}, extended to have s(0) = 0. A perfect number is nothing other than a positive integer fixed point. We say n is amicable if n generates a two-cycle: in other words, s(n) 6= n and s(s(n)) = n. For example, s(220) = 284, ...
... Did Pythagoras invent arithmetic dynamics? Consider the map s : N ∪ {0} → N ∪ {0}, extended to have s(0) = 0. A perfect number is nothing other than a positive integer fixed point. We say n is amicable if n generates a two-cycle: in other words, s(n) 6= n and s(s(n)) = n. For example, s(220) = 284, ...
references
... (i) the least finite number of summands, g(m, r) < , and (ii) the finite invariant set, Z(m, r) , such, that for any s g(m, r), we have N(m, r, s) = {s mr + z : z Z(m, r)}. Theorem 1 is the solution to a rather natural generalization of the Classical Waring Problem (CWP); the latter was sol ...
... (i) the least finite number of summands, g(m, r) < , and (ii) the finite invariant set, Z(m, r) , such, that for any s g(m, r), we have N(m, r, s) = {s mr + z : z Z(m, r)}. Theorem 1 is the solution to a rather natural generalization of the Classical Waring Problem (CWP); the latter was sol ...
DIVISIBILITY PROPERTIES OF CLASS NUMBERS 1. Introduction
... very important to know the structure of class groups Cl(K). However, very little is known. In this paper, we will survey the recent works on this subject, specially, related to quadratic number fields. 2. Imaginary quadratic fields The following theorem on divisibility of class numbers of imaginary ...
... very important to know the structure of class groups Cl(K). However, very little is known. In this paper, we will survey the recent works on this subject, specially, related to quadratic number fields. 2. Imaginary quadratic fields The following theorem on divisibility of class numbers of imaginary ...
On the prime factors of the number 2 p-1 - 1
... prime factor of the number 2 ~ -1 is the second prime factor of the form (/? - l)x +1. The primep which divided 2(p~1)lq— 1 is the third prime factor of the form(/?—l)x+ 1. Now suppose that/? = 5 (mod 8). In view of a theorem of A. Schinzel [8], there exist two primitive prime factors q and r of the ...
... prime factor of the number 2 ~ -1 is the second prime factor of the form (/? - l)x +1. The primep which divided 2(p~1)lq— 1 is the third prime factor of the form(/?—l)x+ 1. Now suppose that/? = 5 (mod 8). In view of a theorem of A. Schinzel [8], there exist two primitive prime factors q and r of the ...
MATH 3527 Number Theory Spring 2016, CRN 35434
... quite challenging. Generally there is no indication of the level of challenge of these exercises nor are there hints. The author appeals to your desire to discover the many beautiful truths of number theory. I hope that you will enjoy the text as much as I do. Other materials: A calculator such as t ...
... quite challenging. Generally there is no indication of the level of challenge of these exercises nor are there hints. The author appeals to your desire to discover the many beautiful truths of number theory. I hope that you will enjoy the text as much as I do. Other materials: A calculator such as t ...
Carom 1-8: Repunits
... (2n)2 = 4n2, and so has remainder 0, while (2n+1)2 = 4n2 + 4n + 1, and so has remainder 1. ...
... (2n)2 = 4n2, and so has remainder 0, while (2n+1)2 = 4n2 + 4n + 1, and so has remainder 1. ...
Full text
... for each integer k>l, one can ask for odd multiperfect numbers, i.e., odd solutions N of the equation
... for each integer k>l, one can ask for odd multiperfect numbers, i.e., odd solutions N of the equation
Transcendence of generalized Euler constants,
... 1. INTRODUCTION. There are two types of numbers: algebraic and transcendental. A complex number is called algebraic if it is a root of some algebraic equation with integer coefficients. A complex number that is not algebraic, is called transcendental. The theory of transcendental numbers arose in co ...
... 1. INTRODUCTION. There are two types of numbers: algebraic and transcendental. A complex number is called algebraic if it is a root of some algebraic equation with integer coefficients. A complex number that is not algebraic, is called transcendental. The theory of transcendental numbers arose in co ...
even, odd, and prime integers
... 13,17, 19 ,31, 61, 89, 107, 127,…..The first 44 Mersenne primes are presently known ...
... 13,17, 19 ,31, 61, 89, 107, 127,…..The first 44 Mersenne primes are presently known ...
Applications of Pell`s Equation
... At n = 1 this is Pythag. triangle with consecutive legs (Appn. 5). Writing z = x − y , some algebra turns it into (y + n(1 − z))2 = n(n + 1)z(z − 1). Let n(n + 1) = a2 b, b squarefree. Then a2 b|(y + n(1 − z))2 , so (!) ab|(y + n(1 − z)). Write y + n(1 − z) = abw , w ∈ Z. Then the displayed equation ...
... At n = 1 this is Pythag. triangle with consecutive legs (Appn. 5). Writing z = x − y , some algebra turns it into (y + n(1 − z))2 = n(n + 1)z(z − 1). Let n(n + 1) = a2 b, b squarefree. Then a2 b|(y + n(1 − z))2 , so (!) ab|(y + n(1 − z)). Write y + n(1 − z) = abw , w ∈ Z. Then the displayed equation ...
c dn> = loglog x + Bl + O(l/log x)
... If we start with (4-l) and reason as above, we might suppose v, - Var, v loglog x. This is quite wrong in view of Theorems 8 and 9. A Poisson distribution with parameter h concentrates its mass close to the mean A so that (4-2) is more relevant than (4-l). Let S be the set of square free numbers. Gi ...
... If we start with (4-l) and reason as above, we might suppose v, - Var, v loglog x. This is quite wrong in view of Theorems 8 and 9. A Poisson distribution with parameter h concentrates its mass close to the mean A so that (4-2) is more relevant than (4-l). Let S be the set of square free numbers. Gi ...
primality proving - American Mathematical Society
... Primality testing is one of the most flourishing fields in computational number theory. Dating back to Gauss, the interest has recently risen with modern cryptology [16]. For quite a long time, it has been known that one could quickly recognize most composite numbers using Fermat's little theorem. F ...
... Primality testing is one of the most flourishing fields in computational number theory. Dating back to Gauss, the interest has recently risen with modern cryptology [16]. For quite a long time, it has been known that one could quickly recognize most composite numbers using Fermat's little theorem. F ...
On the multiplicative properties of arithmetic functions
... J(f, &), the absolute index of /, is defined to be the common value of β(f, &) and β(f, .&), if such a common value exists. When no confusion is possible we will write /(/), J(f), for /(/, &)j J(f, .&), respectively. Occasionally we will talk about /(/, &) or J(f, &) without prefacing our discussion ...
... J(f, &), the absolute index of /, is defined to be the common value of β(f, &) and β(f, .&), if such a common value exists. When no confusion is possible we will write /(/), J(f), for /(/, &)j J(f, .&), respectively. Occasionally we will talk about /(/, &) or J(f, &) without prefacing our discussion ...
ON THE NUMBER OF SPECIAL NUMBERS For lack of a better word
... we have all the non-zero αi are distinct. This concept was introduced by Bernardo Recamán Santos (See mathoverflow [11]). He asked if there is a sequence of 23 consecutive special numbers. This seemingly simple question is still unsolved and it is curious that it is easy to see that there is no seq ...
... we have all the non-zero αi are distinct. This concept was introduced by Bernardo Recamán Santos (See mathoverflow [11]). He asked if there is a sequence of 23 consecutive special numbers. This seemingly simple question is still unsolved and it is curious that it is easy to see that there is no seq ...
When is na member of a Pythagorean Triple?
... Pythagoras' Theorem is perhaps the best known result in the whole of mathematics and yet many things remain unknown (or perhaps just "unstudied") about the consequences of this 'simple' Theorem. In this article we investigate which numbers can be part of triples such as {3, 4, 5} and {5, 12, 13} - r ...
... Pythagoras' Theorem is perhaps the best known result in the whole of mathematics and yet many things remain unknown (or perhaps just "unstudied") about the consequences of this 'simple' Theorem. In this article we investigate which numbers can be part of triples such as {3, 4, 5} and {5, 12, 13} - r ...
For screen
... 1 allows us to prove that the latter equation with a ≥ 2 is not solvable for Q ∈ {2, 3, 5, 7, 13, 17, 19, 37, 73, 97, . . . }. In order to give a precise statement, we need first to introduce some notations. Let S1 be the set of all positive integers greater than 1 and composed only by 2 and by the ...
... 1 allows us to prove that the latter equation with a ≥ 2 is not solvable for Q ∈ {2, 3, 5, 7, 13, 17, 19, 37, 73, 97, . . . }. In order to give a precise statement, we need first to introduce some notations. Let S1 be the set of all positive integers greater than 1 and composed only by 2 and by the ...
Why Cryptography?
... bound x, where x is a real number. For a real number x ¸ 0, the function (x) is defined to be the number of primes up to x. Thus, (1) = 0, (2) = 1, (7.5) = 4, and so on. ...
... bound x, where x is a real number. For a real number x ¸ 0, the function (x) is defined to be the number of primes up to x. Thus, (1) = 0, (2) = 1, (7.5) = 4, and so on. ...