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Register a new userSome conjectures in elementary number theory
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Authors
Angelo B. Mingarelli
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We announce a number of conjectures associated with and arising from a study of primes and irrationals in $mathbb{R}$. All are supported by numerical verification to the extent possible.
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Let $Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $fcolonmathbb{N}tomathbb{C}$ one has [ frac{1}{N}sum_{n=1}^N, f(Omega(n)+1)=frac{1}{N}sum_{n=1}^N, f(Omega(n))+mathrm{o}_{Ntoinfty}(1). ] This yields a new elementary proof of the Prime Number Theorem.
We first formulate a function field version of Vojtas generalized abc conjecture for algebraic tori. We then show a function field analogue of the Lang-Vojta Conjecture for varieties of log general type that are ramified covers of $mathbb G_m^n$. In particular, it includes the case $ mathbb P^nsetminus D$, where $D$ is a hypersurface over a function field in $mathbb P^n$ with $n+1$ irreducible components and $deg Dge n+2$. The main tools include generalizations of the techniques developed by Corvaja and Zannier in 2008 and 2013 and a gcd estimate of two multivariable polynomials over function fields evaluated at $S$-unit arguments. The gcd theorem obtained here is an adaptation of Levins methods for number fields in 2019 via a weaker version of Schmidts subspace theorem over function fields, which we derive with the use of Vojtas machine in a setting over the constant fields.
In this paper we show how the cross-disciplinary transfer of techniques from Dynamical Systems Theory to Number Theory can be a fruitful avenue for research. We illustrate this idea by exploring from a nonlinear and symbolic dynamics viewpoint certain patterns emerging in some residue sequences generated from the prime number sequence. We show that the sequence formed by the residues of the primes modulo $k$ are maximally chaotic and, while lacking forbidden patterns, display a non-trivial spectrum of Renyi entropies which suggest that every block of size $m>1$, while admissible, occurs with different probability. This non-uniform distribution of blocks for $m>1$ contrasts Dirichlets theorem that guarantees equiprobability for $m=1$. We then explore in a similar fashion the sequence of prime gap residues. This sequence is again chaotic (positivity of Kolmogorov-Sinai entropy), however chaos is weaker as we find forbidden patterns for every block of size $m>1$. We relate the onset of these forbidden patterns with the divisibility properties of integers, and estimate the densities of gap block residues via Hardy-Littlewood $k$-tuple conjecture. We use this estimation to argue that the amount of admissible blocks is non-uniformly distributed, what supports the fact that the spectrum of Renyi entropies is again non-trivial in this case. We complete our analysis by applying the Chaos Game to these symbolic sequences, and comparing the IFS attractors found for the experimental sequences with appropriate null models.
Let ${F}_{n}$ be the Farey sequence of order $n$. For $S subseteq {F}_n$ we let $mathcal{Q}(S) = left{x/y:x,yin S, xle y , , textrm{and} , , y eq 0right}$. We show that if $mathcal{Q}(S)subseteq F_n$, then $|S|leq n+1$. Moreover, we prove that in any of the following cases: (1) $mathcal{Q}(S)=F_n$; (2) $mathcal{Q}(S)subseteq F_n$ and $|S|=n+1$, we must have $S = left{0,1,frac{1}{2},ldots,frac{1}{n}right}$ or $S=left{0,1,frac{1}{n},ldots,frac{n-1}{n}right}$ except for $n=4$, where we have an additional set ${0, 1, frac{1}{2}, frac{1}{3}, frac{2}{3}}$ for the second case. Our results are based on Grahams GCD conjectures, which have been proved by Balasubramanian and Soundararajan.
Fix a smooth cubic form $F/mathbb{Q}$ in $6$ variables. For $N_F(X):=#{boldsymbol{x}in[-X,X]^6:F(boldsymbol{x})=0}$, the randomness prediction $N_F(X)=(c_text{HL}+o(1))cdot X^3$ as $Xtoinfty$ of Hardy-Littlewood may fail. Nonetheless, Hooley suggested a modified prediction accounting for special structured loci on the projective variety $V:= V(F)subseteqmathbb{P}^5_mathbb{Q}$. A weighted version of $N_F(X)$ essentially decomposes as a sum of adelic data over hyperplane sections $V_{boldsymbol{c}}subseteq V$, generically with nonzero discriminant $D(boldsymbol{c})$. Assuming standard conjectures for the Hasse-Weil $L$-functions $L(s,V_{boldsymbol{c}})$ over ${boldsymbol{c}inmathbb{Z}^6:D(boldsymbol{c}) eq0}$, Hooley proved the bound $N_F(X) = O_epsilon(X^{3+epsilon})$, essentially for any given diagonal $F$. Now assume (1) standard conjectures for each $L(s,V_{boldsymbol{c}})$, for certain tensor $L$-functions thereof, and for $L(s,V)$; (2) standard predictions (of Random Matrix Theory type) for the mean values of $1/L(s)$ and $1/L(s_1)L(s_2)$ over certain geometric families; (3) a quantitative form of the Square-free Sieve Conjecture for $D$; and (4) an effective bound on the local variation (in $boldsymbol{c}$) of the local factors $L_p(s,V_{boldsymbol{c}})$, in the spirit of Krasners lemma. Under (1)-(4), we establish (away from the Hessian of $F$) a weighted, localized version of Hooleys prediction for diagonal $F$ -- and hence the Hasse principle for $V/mathbb{Q}$. Still under (1)-(4), we conclude that asymptotically $100%$ of integers $a otin{4,5}bmod{9}$ lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}}$ -- and a positive fraction lie in ${x^3+y^3+z^3:x,y,zinmathbb{Z}_{geq0}}$.
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