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If $p$ is a prime of size 5 or greater then it is not good. We can pick a prime $q$ such that $q$ is greater than $p$ and less than $2p-1$. If $p$ is greater than 24 we can find such a number by using …

For any prime $p$ greater than 3 $p^{2}$ is bad. If there is prime $q$ greater than $p^{2}$ and less than $p^{2}+p-1$ then I claim that $q/p^{4}$ cannot be expressed and hence p is bad. Because $q$ is …

If we assume the conjecture that $6$ is the only even number such that $F_{2n}$ is the sum of two squares then $2$ cannot divide $n$. Then we also have $F_{2n}=F_n*L_n$ and so if $F_{2n}$ is the sum …

I think there are bases $b$ where the probability becomes arbitrarily low. Let $b$ be the product of the first $n$ primes plus one then the difference of $b$ and its palindrome will be divisible by $b …

I think that we have had a previous thread related idoneal numbers here. The known idoneal numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48 …

For large enough x, x+x^.525 contains a prime see: R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proceedings of the London Mathematical Society 83, (2001), 532–5 …

If I plug the following polynomial: 3x^3-x into the formula for the discriminant I think I get 12. The formula for the discriminant of the cubic I am using is here: http://en.wikipedia.org/wiki/Discr …

Equation 7 is derived by using the resultant of the two polynomials in 5 and 6. There is a wikipedia article on the resultant. It can be computed using the Sylvester matrix which again has a wikipedia …

It is not known whether pi has a random distribution of digits in its expansion. Here is an article on this and related problems: http://mathworld.wolfram.com/NormalNumber.html

For quantum computers it is in BQP since factoring is in BQP see the wikipedia article on Shor's algorithm. The general number field sieve is the most efficient classical algorithm for factoring numbe …

I don't think we can do this for all k and d. Look at the subset of [1,N] consisting of all points greater then (4/10)N and less then (5/10)N this will have density 1/10 and any half arithmetic progr …

There is a paper online about this here: http://www-math.mit.edu/~green/ramanujanconstant.pdf

If we have a function of radius 1 then by Carlson's theorem as noted above the function is either a rational function or has a natural boundary. For it to be meromorphic it must not have a natural bou …

The paper you have quoted says that if the generalized Riemann hypothesis holds then there are only 65 idoneal numbers(see corollary 23). This agrees with the first comment to your answer. According t …

I found a paper here: http://eprint.iacr.org/2009/008.pdf which generalizes a result from Lenstra's and Pomerance's paper. The paper is "A note on Agrawal conjecture" by Roman Popovych. Here is the …