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Old 2021-08-02, 11:13   #1
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

72·73 Posts
Default Unsolved problems about primes and number theory collect

1. Grand Riemann hypothesis
1a. Extended Riemann hypothesis
1aa. Generalized Riemann hypothesis
4. Mertens conjecture (disproved)
1aaa./4a. Riemann hypothesis
1aab. class number problem for imaginary quadratic number fields, i.e. there is no d>163 such that -d has class number 1 (proved)

2. n conjecture
2a. abc conjecture
2aa. Fermat–Catalan conjecture
2aaa. Beal conjecture
5. Euler's sum of powers conjecture (disproved)
5a. Lander, Parkin, and Selfridge conjecture
2aaaa./5aa. Fermat's Last Theorem (proved)
2ab. Pillai's conjecture
2aab./2aba. Catalan's conjecture (proved)
2b. Szpiro's conjecture

3. Bateman–Horn conjecture
3a. Schinzel's hypothesis H
3aa. Bunyakovsky conjecture
3aaa. Landau 4th problem
3ab. Dickson's conjecture
3aba. Hardy–Littlewood 1st conjecture
3b. PNT in AP (proved)
3aab./3abb./3ba. Dirichlet's theorem on arithmetic progressions (proved)
3abc./3bb. length of primes in arithmetic progression has no upper bound (proved)
3abd. length of (1st or 2nd) Cunningham chain has no upper bound
3abaa. Polignac's conjecture
3abab. Hardy–Littlewood 2nd conjecture is false
3abaaa. Twin prime conjecture
3abaaaa. There are infinitely many Chen primes (proved)

(e.g. conjecture * covers conjecture *a, *b, *c, ..., where * is any string, tell me if I miss any conjectures in these families)

These 1 (Grand Riemann hypothesis, stronger than Riemann hypothesis), 2 (n conjecture, stronger than abc conjecture), and 3 (Bateman–Horn conjecture, stronger than Schinzel's hypothesis H), I call these three conjectures "three classed unsolved problems in number theory" and hope that they are all true, at least 1' (Riemann hypothesis), 2' (abc conjecture), and 3' (Schinzel's hypothesis H) are all true.

Last fiddled with by sweety439 on 2021-08-05 at 05:48
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Old 2022-03-11, 23:25   #2
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

72·73 Posts

Like Grand Riemann hypothesis, n conjecture, Bateman–Horn conjecture, I like the conjecture including as many conjectures as possible, e.g.

A085398 (instead of its subsequence), including A066180 (prime n), A103795 (n=2*p with odd prime p), A056993 (n=2^i with i>=1), A153438 (n=3^i with i>=2), A246120 (n=2*3^i with i>=1), A246119 (n=2^i*3 with i>=1), A298206 (n=2^i*3^2 with i>=1), A246121 (n=2^i*3^i with i>=1), A206418 (n=5^i with i>=2), A205506 (n=2^i*3^j with i,j>=1), A181980 (n=2^i*5^j with i,j>=1).

Smallest totient number k > 1 such that n*k is a nontotient number, or 0 if no such number exists (instead of its subsequences A350085 (n in A007617) and A350086 (n in A005277))

A326615 (instead of A306499, there is a similar extension for A306500, but this sequence is currently not in OEIS)

A309129 (instead of A000926 and A003173, note that there are some numbers in A309129 which are in neither A000926 nor in A003173, such as 247 and 267 (also 1467, but 1467 = 3^2 * 163, thus it is equivalent to 163) (also a similar sequence for +n (instead of -n) is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n, the largest such n is (conjectured to be) 1722, but this sequence is currently not in OEIS)

A347567 and A347568 (instead of A020495, A065377, A060003, A042978, A065397, A255904) (twice square numbers and twice triangular numbers are excluded since they will make infinite families, thus especially not A064233, A014090, A076768, A111908)

A039951 (composite n, nonsquarefree n, perfect power n, are not excluded)

A247093 and proposed A326653 (instead of their subsequences A128164, A125713, A084742, A247244)

A328497 (also has the terms 16843^4 and 2124679^4, see A088164) instead of A228562 and A267824 (also not A082180 and A136327, since they include infinite families (the primes and the square of the primes and the cube of the primes)

Extended Sierpinski and Riesel problems (instead of the original Sierpinski and Riesel problems, extended these conjectures to the k such that k+-1 is not coprime to b-1, also GFN, half GFN, GRU, are not excluded)

The minimal prime (start with b+1) problem (including finding the smallest prime of these form for fixed base b: (b^n-1)/(b-1), b^n+1, (b^n+1)/2 (n>=2), k*b^n+1 for all k<b, k*b^n-1 for all k<b, b^n+k for all k<b, b^n-k (n>=2) for all k<b)

But should not including repeating problems (e.g. A061653(2*n) (which is currently not in OEIS, but a similar sequence (I do not like this sequence, since this sequence does not satisfying the condition that the n should be sorted) A061494) rather than A061653(n) (also not A000927(n), since composite n should not be excluded)

Last fiddled with by sweety439 on 2022-03-13 at 08:03
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