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Results tagged with prime-numbers
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user 35626
Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.
3
votes
Divergence of a series similar to $\sum\frac{1}{p}$
The question makes sense if the $r$'s are all positive. The answer is likely to be yes for
the folllowing reason: if it were no, there would be an explicit sequence where the
$r$'s grow faster than $ …
- 13.4k
2
votes
Distribution of composite numbers
The result is wrong. The major problem is that there is little control over the
common elements of the sets A_i. I suggest coming up with simpler sets of
conditions and checking relations between th …
- 2,158
2
votes
Conjecture on the square root of the sum of the squares of the prime factors of a number
It may be of interest to consider in general when A=A_n is integral. I will assume n is given and
drop the subscript.
A is integral when n =p^k, for p prime and k a square.
A integral and n composi …
- 2,158
0
votes
A prime sequence can be partitioned into two sets of equal or consecutive sum
Expanding on the comment above, consider Pn, the set of the first n primes,
and SSn, the set of subset sums of Pn. For n greater than 3, we see that
SSn is 6 numbers shy of being the interval [0, Sn], …
- 2,158
2
votes
The shortest interval for which the prime number theorem holds
The short answer is no. It is likely to be sufficient, although right now the best known is
actually that pi(x + x^{0.525}) > pi(x) for all large x (and likely all x > 117). If it were necessary,
thi …
- 2,158
16
votes
Accepted
Arbitrarily long arithmetic progressions
A simple proof is available as well. Pick p coprime to d and let t be such that td=1 mod p. Then, mod p, t times the arithmetic progression looks like a sequence of consecutive integers. Thus its l …
- 2,158
2
votes
1
answer
532
views
On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski
In response to a comment posted under
Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd prim …
CommunityBot
- 1
1
vote
Is there a lower bound for the first non-trivial sequence of consecutive integers where each...
Your conditions seem to imply a search for a confluence of a sizable prime gap in which a
not very smooth number (one with least prime factor of $p_n$) occurs. You can limit the search
by looking "be …
- 2,158