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Register a new userSolutions of $phi(n)=phi(n+k)$ and $sigma(n)=sigma(n+k)$
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Authors
Kevin Ford
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We show that for some $kle 3570$ and all $k$ with $442720643463713815200|k$, the equation $phi(n)=phi(n+k)$ has infinitely many solutions $n$, where $phi$ is Eulers totient function. We also show that for a positive proportion of all $k$, the equation $sigma(n)=sigma(n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.
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The appearance of some papers dealing with the $K^- d to pi Sigma n$ reaction, with some discrepancies in the results and a proposal to measure the reaction at forward $n$ angles at J-PARC justifies to retake the theoretical study with high precision to make accurate predictions for the experiment and extract from there the relevant physical information. We do this in the present paper showing results using the Watson approach and the truncated Faddeev approach. We argue that the Watson approach is more suitable to study the reaction because it takes into account the potential energy of the nucleons forming the deuteron, which is neglected in the truncated Faddeev approach. Predictions for the experiment are done as well as spectra with the integrated neutron angle.
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In the present work, we investigate the hidden-strangeness production process in the $S=+1$ channel via $K^+pto K^+phi,p$, focussing on the exotic textit{pentaquark} molecular $K^*Sigma$ bound state, assigned by $P^+_s(2071,3/2^-)$. For this purpose, we employ the effective Lagrangian approach in the tree-level Born approximation. Using the experimental and theoretical inputs for the exotic state and for the ground-state hadron interactions, the numerical results show a small but obvious peak structure from $P^+_s$ with the signal-to-background ratio $approx1.7,%$, and it is enhanced in the backward-scattering region of the outgoing $K^+$ in the center-of-mass frame. We also find that the contribution from the $K^*(1680,1^-)$ meson plays an important role to reproduce the data. The proton-spin polarizations are taken into account to find a way to reduce the background. The effects of the possible $27$-plet pentaquark $Theta^{++}_{27}$ is discussed as well.
The formation of a deeply-bound $K^- pp$ state with $I=1/2$, $J^pi=0^-$ by the $^3$He(in-flight $K^-$, $n$) reaction is theoretically investigated in a distorted-wave impulse approximation using the Greens function method. The expected inclusive and semi-exclusive spectra at $p_{K^-} = 1.0$ GeV/c and $theta_{rm lab} = 0^{circ}$ are calculated for the forthcoming J-PARC E15 experiment. We demonstrate these spectra with several types of phenomenological $K^-$-``$pp$ optical potentials $U^{rm opt}(E)$ which have an energy-dependent imaginary part multiplied by a phase space suppression factor, fitting to recent theoretical predictions or experimental candidates of the $K^-pp$ bound state. The results show that a cusp-like peak at the $pi Sigma N$ threshold is an unique signal for the $K^-pp$ bound state in the spectrum including the [$K^-pp$] $to$ $Y + N$ decay process from the two-nucleon $K^-$ absorption, as well as a distinct peak of the $K^-pp$ bound state. The shape of the spectrum is explained by a trajectory of a moving pole of the $K^-pp$ bound state in the complex energy plane. The importance of the spectrum with [$K^-pp$] $to$ $Y + N$ from the two-nucleon $K^-$ absorption is emphasized in order to extract clear evidence of the $K^-pp$ bound state.
We investigate the $phi$ meson nuclear transparency using some recent theoretical developments on the $phi$ in medium self-energy. The inclusion of direct resonant $phi N$-scattering and the kaon decay mechanisms leads to a $phi$ width much larger than in most previous theoretical approaches. The model has been confronted with photoproduction data from CLAS and LEPS and the recent proton induced $phi$ production from COSY finding an overall good agreement. The results support the need of a quite large direct $phi N$-scattering contribution to the self-energy.
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