This paper proves Birch and Swinnerton-Dyer conjecture for Q-curves defined over real quadratic number field with everywhere good reductions.

This paper is devoted to finding the highest possible focus order of planar polynomial differential equations. The results consist of two parts: (i) we explicitly construct a class of concrete systems of degree n, where n+1 is a prime p or a power of a prime p^k, and show that these systems can have a focus order n^2-n; (ii) we theoretically prove the existence of polynomial systems of degree n having a focus order n^2-1 for any even number n. Corresponding results for odd n and more concrete examples having higher focus orders are given too.

No abstract uploaded!

We introduce the prime density triangle, a multiplicative array constructed by a modified rule of indices. Two display formats are given: a right-angle triangle with converted and unconverted entries, and an equilateral triangle obtained by mirroring the right-angle triangle about the central term xn. We derive the row-sum generat- ing function and provide worked examples using both direct computation and the generating function. Using a four-rule division scheme applied to the unconverted rows we obtain the sequence whose integer parts sum to the magic numbers for two spin orientations. Replacing the repeated end integers by 1,1 yields integer parts that give the standard harmonic oscillator (HO) magic numbers and, in conjunction with the two-spin-orientation nucleon magic numbers, gives precisely the standard nuclear magic numbers 2, 8, 20, 28, 50, 82, 126, 184. Furthermore, digital-root reduc- tion of the converted triangle reveals a distinct “4,9,9” pattern whose diagonals concatenate into large primes, establishing a direct arithmetic link between prime distribution and the symmetry conditions of nuclear shell closures. Subshell filling up to x9, deductions, and evidence from nuclear physics are included. A recurrence M = m ± [(n1n2) + 2] is presented for generating further magic numbers.The sequences appear as OEIS A005897, OEIS A018226 and OEIS A007290.

No abstract uploaded!