Add to ORKGEdmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let β=(((24!)!)!)!, and let Φ denote the implication: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆(-∞,β]. We heuristically justify the statement Φ without invoking Landau's conjecture. The set X = {k∈N: (β<k) ⇒ (β,k)∩P(n^2+1) ≠ ∅} satisfies conditions (1)-(4). (1) There are a large number of elements of X and it is conjectured that X is infinite. (2) No known algorithm decides the finiteness/infiniteness of X . (3) There is a known algorithm that for every n∈N decides whether or not n ∈ X. (4) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n]. (5) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n] and some known definition...
This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on d...
In this paper, we show under the abc conjecture that the Diophantine equation f(x)=u!+v! has only fi...
For a set of primes P, let Ψ(x;P) be the number of positive integers n≤x all of whose prime factors ...
Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
Let b=((24!)!)!, and let P_{n^2+1} denote the set of all primes of the form n^2+1. Let M denote the ...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
Let \beta=((24!)!)!, and P_{n^2+1} denote the set of all primes of the form n^2+1. Let M denote the ...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
We assume that the current mathematical knowledge K is a finite set of statements from both formal a...
Conditions (1)-(8) below concern sets X \subseteq N. (1) There are a large number of elements of X a...
After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive pri...
This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on d...
This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on d...
In this paper, we show under the abc conjecture that the Diophantine equation f(x)=u!+v! has only fi...
For a set of primes P, let Ψ(x;P) be the number of positive integers n≤x all of whose prime factors ...
Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
Let b=((24!)!)!, and let P_{n^2+1} denote the set of all primes of the form n^2+1. Let M denote the ...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
Let \beta=((24!)!)!, and P_{n^2+1} denote the set of all primes of the form n^2+1. Let M denote the ...
Let f(1)=2, f(2)=4, and let f(n+1)=f(n)! for every integer n≥2. Edmund Landau's conjecture states th...
We assume that the current mathematical knowledge K is a finite set of statements from both formal a...
Conditions (1)-(8) below concern sets X \subseteq N. (1) There are a large number of elements of X a...
After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive pri...
This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on d...
This is an expanded and revised version of the article: A. Tyszka, Statements and open problems on d...
In this paper, we show under the abc conjecture that the Diophantine equation f(x)=u!+v! has only fi...
For a set of primes P, let Ψ(x;P) be the number of positive integers n≤x all of whose prime factors ...