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Mathguy15
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a and b are real numbers such that the sequence{c}n=1--->{infinity} defined by c_n=a^n-b^n contains only integers. Prove that a and b are integers.
Mathguy
Mathguy
Mathguy15 said:a and b are real numbers such that the sequence{c}n=1--->{infinity}
defined by c_n=a^n-b^n contains only integers. Prove that a and b are integers.
Mathguy
Norwegian said:a+b is rational, and we get a and b are rational.
Dodo said:Sorry, Norwegian, but why? For example, sqrt(2) and 3-sqrt(2) are both irrational, and they add up to 3.
checkitagain said:[itex]c_n \ = \ a^n - b^n[/itex]
What about any real numbers a and b, such that a = b, so that [itex]c_n = 0 ?[/itex]
Here, and b don't have to be integers.
Do I have your problem understood, and/or
are there more restrictions on a and b?
Norwegian said:I assume you mean a≠b.
Since a-b and a2-b2=(a-b)(a+b) are both integers, a+b is rational, and we get a and b are rational.
We can write b=m/t and a=(m+kt)/t with (m,t)=1. Assume t≠1, then there is an integer s such that k is divisible by ts but not by ts+1.
Let p be a prime larger than t and 2s+2.
cp=ap-bp=(pktmp-1+k2t2(...))/tp
Both the second term and the denominator are divisible by t2s+2, while the first term is not, so the fraction is not an integer. It follows that t=1 and we are done.
The Riemann Hypothesis is a conjecture in number theory that states that all non-trivial zeros of the Riemann zeta function lie on the line Re(s) = 1/2. It is important because it has far-reaching implications in number theory, including the distribution of prime numbers. It has been one of the most famous unsolved problems in mathematics for over 150 years.
Prime numbers are essential in modern cryptography, specifically in asymmetric encryption algorithms such as RSA. This is because it is computationally difficult to factorize large numbers into their prime factors, making it a secure way to encrypt information.
The Goldbach Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. It has not been proven yet, but it has been verified for all integers up to 4 x 10^18. It remains one of the oldest and most famous unsolved problems in number theory.
The Collatz Conjecture, also known as the 3n + 1 problem, states that starting from any positive integer, if you repeatedly apply the rules n/2 (if n is even) or 3n + 1 (if n is odd), the sequence will eventually reach 1. Despite its simple formulation, it has been an open problem since 1937 and has been verified for all integers up to 2^60.
A perfect number is a positive integer that is equal to the sum of its proper divisors (i.e. divisors excluding itself). Interestingly, it has been proven that all even perfect numbers are related to Mersenne primes, which are prime numbers of the form 2^n - 1. However, it is not known if there are any odd perfect numbers.