My conjectures in number theory

[sweety439]

Conjecture 1: A generalization of Fermat–Catalan conjecture and of Lander, Parkin, and Selfridge conjecture:

a1^b1+a2^b2+a3^b3+…+am^bm = c1^d1+c2^d2+c3^d3+…+cn^dn

has only finitely many solutions with m, n, ai, bi, cj, dj all positive integers, gcd(a1,a2,a3,…,am,c1,c2,c3,…,cn) = 1, sigma(A) != sigma(C) for all nonempty proper subset A of {ai^bi, i = 1, 2, ..., m} and all nonempty proper subset C of {cj^dj, j = 1, 2, ..., n}, 1/b1+1/b2+1/b3+…+1/bm+1/d1+1/d2+…+1/dn < 1

If we require m+n = 3, then this is Fermat–Catalan conjecture.
If we require b1=b2=b3=…=bm=d1=d2=d3=…=dn, then this is Lander, Parkin, and Selfridge conjecture.

Conjecture 2: A generalization of Schinzel’s hypothesis H to exponential sequences (a*b^n+c)/gcd(a+c,b-1) (with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1, and variable integer n>=1)

If (a*b^n+c)/gcd(a+c,b-1) (with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1, and variable integer n>=1) cannot be proven to only contain composites or to only contain finitely many primes, by using covering congruence, algebraic factorization, or combine of them, then (a*b^n+c)/gcd(a+c,b-1) (with fixed integers a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1, and variable integer n>=1) contains infinitely many primes.

Conjecture 3: A generalization of Catalan’s Aliquot sequence to quasi-Aliquot sequences (like quasiperfect numbers and quasi-amicable numbers):

The quasi-Aliquot sequences for any nonnegative integer must terminate at 0 or a quasi-sociable numbers cycle with even length, also there is no quasi-sociable numbers cycle with odd (including 1) length.

Conjecture 4: There are only finitely many numbers n not == 0 or +-1 mod 12 such that omega((n-1)*n*(n+1)) <= 4, and the largest of them is 3^541-1

Conjecture 5: All bases b > 4712 has a strong pseudoprime < b-1

Conjecture 6: There is no composite which is strong pseudoprime to both base 2 and base 3 and strong pseudoprime with parameters (P, Q) defined by Selfridge's Method A

Conjecture 7: Phi(n,b)/gcd(Phi(n,b)) is never perfect power if n > 2 except (n,b) = (3,18), (5,3), (6,19), (3,A028231), (4,A002315), (6,A028231+1)

Conjecture 8: For n >= 2, there is always a twin prime pair between n and 3*n

Conjecture 9: All numbers n > 619 contains at least one digit 0 in some base 3<=b<=n-1, and all numbers n > 256 contains at least one digit b-1 in some base 3<=b<=n

Conjecture 10: There is only finitely many n coprime to 6 and cannot be written as x+y, where x is in A210479 and y is in A258838

[MattcAnderson]

All possible
Have a nice day

In my world, conjecture is like probably true