Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

 

Let three positive integers $a, b, c$ with $\gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

 

Stronger conjecture: Let two positive integers $a, b$ with $\gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Note that the stronger conjecture is true up to $10^{18}$ (by Yaakov Baruch)

Relative question:

• Is min exponents of three positive integers n , n+1 and n+2 =1 true or false?

See also:

• Six Exponentials Theorem

• Four Exponentials Conjecture

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

Let three positive integers $a, b, c$ with $\gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

Stronger conjecture: Let two positive integers $a, b$ with $\gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Note that the stronger conjecture is true up to $10^{18}$ (by Yaakov Baruch)

Relative question:

• Is min exponents of three positive integers n , n+1 and n+2 =1 true or false?

See also:

• Six Exponentials Theorem

• Four Exponentials Conjecture

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

Let three positive integers $a, b, c$ with $gcd(a,b)$$=gcd(b,c)$$=gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

Stronger conjecture: Let two positive integers $a, b$ with $gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Note that the stronger conjecture is true up to $10^{18}$ (by Yaakov Baruch)

Relative question:

• Is min exponents of three positive integers n , n+1 and n+2 =1 true or false?

See also:

• Six Exponentials Theorem

• Four Exponentials Conjecture

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

Let three positive integers $a, b, c$ with $\gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

Stronger conjecture: Let two positive integers $a, b$ with $\gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Note that the stronger conjecture is true up to $10^{18}$ (by Yaakov Baruch)

Relative question:

• Is min exponents of three positive integers n , n+1 and n+2 =1 true or false?

See also:

• Six Exponentials Theorem

• Four Exponentials Conjecture

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

Let three positive integers $a, b, c$ with $gcd(a,b)$$=gcd(b,c)$$=gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

Stronger conjecture: Let two positive integers $a, b$ with $gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Relative question:

• Is min exponents of three positive integers n , n+1 and n+2 =1 true or false?

See also:

• Six Exponentials Theorem

• Four Exponentials Conjecture

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

Let three positive integers $a, b, c$ with $gcd(a,b)$$=gcd(b,c)$$=gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

Stronger conjecture: Let two positive integers $a, b$ with $gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Note that the stronger conjecture is true up to $10^{18}$ (by Yaakov Baruch)

Relative question:

• Is min exponents of three positive integers n , n+1 and n+2 =1 true or false?

See also:

• Six Exponentials Theorem

• Four Exponentials Conjecture