New in June 4, 2017: The Hardy-Littewood prime race:
A few parts appearing in this excursions to primes:
• We formulate Goldbach conjectures or questions in division algebras and Eisenstein integers.
Gauss Quaternion Eisenstein
Every even Gaussian integer a+ib with a,b larger than 1 is the sum of two Gaussian primes with positive coefficients. Every Lipschitz quaternion with entries larger than 1 is the sum of two Hurwitz quaternions with positive coefficients. Every Eisenstein integer a+b w with a,b larger than 3 is the sum of two Eisenstein primes with positive coefficients.
3+6i=(1+i)+(2+5i) (4+2i+j+2k)=(3+i+j+k)/2+(5+3i+j+3k)/25+2w = (2+w) + (3+w)
In the Gaussian and Quaternion case, we can relate to Landau or Bunyakowsky type conjectures. In the Octonion case, the statement is false for Klainian primes but looks reasonable for Kirmse primes. In the Quaternion and Eisenstein case, no evenness is necessary.
• We look at the zeta function in each of the division algebras and relate to the usual zeta function. We show pictures of the roots of the zeta function for Gaussian integers.
• We illustrate the Gaussian, Hurwitz and Kirmse primes with pictures.
• We look at almost periodic matrices which are the real part of van der Monde matrices. We can estimate the determinant in special Diophantine cases but admire mostly the beautiful snow flake spectra. We have looked at these spectra already in 2008 and used it in computer algebra projects in 2010.
• Gaussian integers define a graph, where two positive integers a,b are connected, if a+ib is prime. The growth rate of the Euler characteristic of this graph is related to the growth of primes.
• We look at greatest common divisor matrices for which the determinants are explicitly known. The spectra in the complex have a spiral shape feature. These matrices were used heavily in our 2015 linear algebra course.
• We look at matrices defined by Gaussian integers. We observe that they become invertible if large enough and some symmetry is avoided. We investigate the growth of the determinant, trace, coefficients of the characteristic polynomials. While writing the exam for Math 21b in the spring 2016, Gaussian matrices were used as an inspiration.
This project was done from April 27 (writing the 21b exam), 2016 to June 19, 2016 (start of summer school). It got started while reading the book of Mazur and Stein. (See my Amazon review of June 27, 2016.)