Generalization of Legendre's conjecture
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I suggest the following is true. I've verified it for small values with Mathematica and heuristically it seems solid.
For $n ge p^k$ with prime $p ge 5$, there are primes $r_i > p$ such that
$$n^2 < r_1 r_2 cdots r_k < (n+1)^2$$
(The requirement $p ge 5$ can probably be swapped for "given large $n$".)
When $k=1$, this is basically equivalent to Legendre's conjecture. When $k=2$, this would address semiprimes. For example, let $n=5^2$, and we predict there will be a semiprime between 625 and 676 whose two factors are at least 7.
The semiprime case also implies that for any $n$, there is a prime $p$ in $n^3 < p < n^3 + 2n + 1$.
Asymptotically, this predicts that there will always be a number between consecutive squares that factors as $k$ primes of a certain size.
I am curious whether or not this is the same as an existing conjecture, or whether anyone sees any obvious flaws with it. I don't think it is directly implied by Legendre's conjecture, but cannot be certain.
number-theory conjectures
asked Aug 14 at 10:14
Trev
185
add a comment |Â
up vote
0
down vote
favorite
I suggest the following is true. I've verified it for small values with Mathematica and heuristically it seems solid.
For $n ge p^k$ with prime $p ge 5$, there are primes $r_i > p$ such that
$$n^2 < r_1 r_2 cdots r_k < (n+1)^2$$
(The requirement $p ge 5$ can probably be swapped for "given large $n$".)
When $k=1$, this is basically equivalent to Legendre's conjecture. When $k=2$, this would address semiprimes. For example, let $n=5^2$, and we predict there will be a semiprime between 625 and 676 whose two factors are at least 7.
The semiprime case also implies that for any $n$, there is a prime $p$ in $n^3 < p < n^3 + 2n + 1$.
Asymptotically, this predicts that there will always be a number between consecutive squares that factors as $k$ primes of a certain size.
I am curious whether or not this is the same as an existing conjecture, or whether anyone sees any obvious flaws with it. I don't think it is directly implied by Legendre's conjecture, but cannot be certain.
number-theory conjectures
asked Aug 14 at 10:14
Trev
185
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I suggest the following is true. I've verified it for small values with Mathematica and heuristically it seems solid.
For $n ge p^k$ with prime $p ge 5$, there are primes $r_i > p$ such that
$$n^2 < r_1 r_2 cdots r_k < (n+1)^2$$
(The requirement $p ge 5$ can probably be swapped for "given large $n$".)
When $k=1$, this is basically equivalent to Legendre's conjecture. When $k=2$, this would address semiprimes. For example, let $n=5^2$, and we predict there will be a semiprime between 625 and 676 whose two factors are at least 7.
The semiprime case also implies that for any $n$, there is a prime $p$ in $n^3 < p < n^3 + 2n + 1$.
Asymptotically, this predicts that there will always be a number between consecutive squares that factors as $k$ primes of a certain size.
I am curious whether or not this is the same as an existing conjecture, or whether anyone sees any obvious flaws with it. I don't think it is directly implied by Legendre's conjecture, but cannot be certain.
number-theory conjectures
asked Aug 14 at 10:14
Trev
185
I suggest the following is true. I've verified it for small values with Mathematica and heuristically it seems solid.
For $n ge p^k$ with prime $p ge 5$, there are primes $r_i > p$ such that
$$n^2 < r_1 r_2 cdots r_k < (n+1)^2$$
(The requirement $p ge 5$ can probably be swapped for "given large $n$".)
When $k=1$, this is basically equivalent to Legendre's conjecture. When $k=2$, this would address semiprimes. For example, let $n=5^2$, and we predict there will be a semiprime between 625 and 676 whose two factors are at least 7.
The semiprime case also implies that for any $n$, there is a prime $p$ in $n^3 < p < n^3 + 2n + 1$.
Asymptotically, this predicts that there will always be a number between consecutive squares that factors as $k$ primes of a certain size.
I am curious whether or not this is the same as an existing conjecture, or whether anyone sees any obvious flaws with it. I don't think it is directly implied by Legendre's conjecture, but cannot be certain.
number-theory conjectures
asked Aug 14 at 10:14
Trev
185
asked Aug 14 at 10:14
Trev
185
asked Aug 14 at 10:14
Trev
185
asked Aug 14 at 10:14
Trev
185
185
add a comment |Â
add a comment |Â
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