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The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparison, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparison by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output, it is a line written in Pari/GP, showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>prime>(log(n))*prime(n)^(1/n)+log(log(n)),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.

The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparison, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparison by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>prime(n)^(1/n),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.

The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparison, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparison by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output, it is a line written in Pari/GP, showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>(log(n))*prime(n)^(1/n)+log(log(n)),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.

Grammar
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user142929
user142929
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An On an attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means

The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparasioncomparison, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparsioncomparison by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>prime(n)^(1/n),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.

An attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means

The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparasion, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparsion by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>prime(n)^(1/n),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.

On an attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means

The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparison, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparison by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>prime(n)^(1/n),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.

Source Link
user142929
user142929
  • 1
  • 1
  • 7
  • 30

An attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means

The idea of this post arises when I've considered simple variants of the known as Firoozbakht's conjecture (see this corresponding Wikipedia or [1]), and comparisons by trial and error with means (if you are interested feel free to do yourself comparisons motivated from the Wikipedia Generalized mean).

I've considered the variant, that is a comparasion, $$\frac{n+p_{n+1}}{n+1}>\lambda(n)\cdot p_{n}^{\frac{1}{n}}+\mu(n)\tag{1}$$ where $p_k$ denotes the $k$-th prime number and $\lambda(n)$ and $\mu(n)$ are simple functions. As I've said in the introductory paragraph I got my comparsion by trial and error using a Pari/GP program, from these calculations I've considered $\lambda(n)=\log n$ and $\mu(n)=\log\log n$, and my proposal (below in my Question) will be for integers $n\geq 2700$.

The identity $(1)$ evokes a comparison for means because the quantity $\frac{n\cdot 1+p_{n+1}}{n+1}$ is the arithmetic mean of $n$ ones $1$'s and $p_{n+1}$ while that $p_{n}^{1/n}$ is the geometric mean of $n-1$ ones $1$'s and the prime $p_n$. The functions $\log n$ and $\log\log n$ are simple functions that additionally belong to a class of functions studied in Rafael Jakimczuk, Functions of Slow Increase and Integer Sequences, Journal of Integer Sequences, Article 10.1.1, Vol. 13 (2010).

Question. Prove or refute the following conjecture as a variant of Firoozbakht's conjecture:

For each integer $n\geq 2700$ the following inequality $$\frac{n+p_{n+1}}{n+1}>p_{n}^{\frac{1}{n}}(\log n)+\log\log n\tag{2}$$ holds.

Many thanks.

My proposal $(2)$ is true for integers $2700\leq n\leq 50000$. You can see it from the web Sage Cell Server using this Pari/GP program (choose GP as language)

for(n=2700, 50000, if((n+prime(n+1))/(n+1)<(log(n))*prime(n)^(1/n)+log(log(n)),print(n)))

(you need to wait a minute to see that there aren't counterexamples as outputs of this program), or this other output showing the differences of LHS and RHS

for(n=2700, 50000, if((n+prime(n+1))/(n+1)>prime(n)^(1/n),print((n+prime(n+1))/(n+1)-(log(n))*prime(n)^(1/n)-log(log(n)))))

As was said feel free to do yourself variants in your home if you can to create a more interesting variant of Firoozbakht's conjecture using generalized means.

References:

[1] Conjecture 30. The Firoozbakht Conjecture, Retrieved 22 August 2012 in Carlos Rivera's web The Prime Puzzles & Problems Connection.