ფაილების კატალოგი

A function ndpf(a) mean number of the distinct prime factors of a and
ndpf(a)=log[2, τ(rad(a))].

Definition. For a positive integer n, the poi of n, denoted poi(n), is the product of the indicators in the canonical disintegration with the prime divisors of the n.
For example:
poi(1)=1,
poi(2^4)=4,
poi(2*5*13)=1,
poi(2^3*5^12*23^6)=3*12*6,
poi(2^4*7^4)=4*4.

Conjecture 1. (Papava). If a+b=c>2 and gcd(a,b)=1 then
c<rad(abc)poi(abc).

Conjecture 2. (Papava). If a+b=c and gcd(a,b)=1 then
poi(abc)≤rad(abc).

A PRIME PROOF OF THE ABC CONJECTURE (Papava)

If a,b,c>4 positive integers, gcd(a,b)=1 and a+b=c then
0<√c<a<c/2<b<c or 0<a<√c<c/2<b<c;
if rad(abc)>√c then ABC conjecture is true;
If rad(abc)<√c<a then ABC conjecture is true because rad(abc)<a<b<c such triples are finite;
if rad(abc)<a<√c then also ABC conjecture is true;
if a<rad(abc)<√c then ...

Let gcd(a,b)=1, a+b=c,
rad(abc)<a<b.
a=?, b=?, c=?.
such triples are infinite.

Conjecture. If x>1,y>1,z,n,m,k∈N, gcd(x,y)=1 and
x^n+y^m=z^k then nmk≤xyz.

A discovery #299
Definition. pqi(n) denote the product of the quality indicators of the distinct prime factors of the integer n.
pqi(3^5*7^2*19^12)=5*2*12=120,
pqi(3^2*5^26)=2*26=52,
pqi(1)=1.

Conjecture (Papava). If gcd(a, b)=1 and a+b=c then
pqi(abc) ≤rad(abc).