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Results tagged with prime-numbers
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user 60732
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
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1
answer
417
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Primes from arithmetic and geometric progressions
The five primes, 131, 157, 211, 349, 739, are neither in arithmetic or geometric progression, but are instead the sum of the five corresponding terms of an arithmetic and geometric progression.
Are …
- 3,707
5
votes
2
answers
309
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Pairs of integers whose product is one more or less than a prime
Given a positive integer N it is often possible to pair each of the integers 1, 2, 3, ..., N with a different integer between N + 1 and 2 N so that the product of each pair is one less or more than a …
- 3,707
2
votes
1
answer
740
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Is every prime greater than 5, less than the sum of the two previous primes?
Can one prove by elementary means (such as Paul Erdös' proof of Bertrand's Postulate) that every prime greater than 5 is less than the sum of the two primes immediately preceding it?
- 3,707
5
votes
2
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483
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Least (and largest) possible number of non-relatively prime pairs among consecutive integers
Given a set of positive integers consider the graph whose vertices are those integers, two of which are joined by an edge if and only if they have a common divisor greater than 1 (i.e, they are not re …
- 3,707
8
votes
4
answers
591
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Covering the primes with pairs of consecutive integers
Is it true that for every sufficiently large positive integer $n$, one can always find at most $k=\lfloor\pi(n)/2\rfloor$ integers, $a_1,a_2,a_3,a_3,\dots a_k$, between $1$ and $n$, such that each of …
- 3,707
3
votes
0
answers
84
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Are numbers which are the product of n primes more common than numbers which are the product... [duplicate]
In a recent video (https://www.facebook.com/188916357807416/videos/519169035700435/) Stephen Wolfram wonders whether, for every integer n>2, eventually the number of integers which are precisely the p …
- 3,707
16
votes
3
answers
2k
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Are 0 and 1, respectively, the least and most used digits among primes?
In order to write the first 25 primes (2 to 97), 46 digits are necessary, nine of each of the digits 2, 3, and 7, fewer of all the others. Thereafter, at least for a while, the digit 1 is used more ti …
- 3,707
11
votes
1
answer
466
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Perfect Runs of Consecutive Integers
A run of 2 or more consecutive integers is said to be perfect if the sum of its terms equals the sum of all the their proper divisors, adding common divisors as often as they occur. For example, 672, …
- 3,707
2
votes
1
answer
174
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Graceful graphs all of whose vertices are labelled with primes or squares
Do graceful graphs exist with more than any arbitrarily large number of vertices, all of which are labelled with a prime or non-negative square number.
Recall that a graceful graph is a graph with m …
- 3,707
7
votes
0
answers
545
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Is every integer greater than 1 the sum of a palindrome and a prime?
Helfgott proved that any odd number greater than 5 is the sum of three primes. Cilleruelo and Luca proved that every positive integer is the sum of three palindromes.
Is every integer greater than 1 …
- 3,707