Journal of Prime Research in Mathematics

Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing four issues in March, June, September and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.

Latest Published Articles

Boyd indices for quasi-normed rearrangement invariant spaces

JPRM-Vol. 1 (2012), Issue 1, pp. 36 – 44 Open Access Full-Text PDF
Abstract: We calculate the Boyd indices for the sum and intersection of two quasi-normed rearrangement invariant spaces. An application to Lorentz type spaces is also given.

Gaps in binary expansions of some arithmetic functions and the irrationality of the Euler constant

JPRM-Vol. 1 (2012), Issue 1, pp. 28 – 35 Open Access Full-Text PDF
Jorge Jimenez Urroz , Florian Luca, Michel Waldschmidt
Abstract: We show that if $$F_n = 22^n+ 1$$ is the nth Fermat number, then the binary digit sum of $$π(F_n)$$ tends to infinity with $$n$$, where $$π(x)$$ is the counting function of the primes $$p ≤ x$$. We also show that if $$F_n$$ is not prime, then the binary expansion of $$φ(F_n)$$ starts with a long string of 1’s, where $$φ$$ is the Euler function. We also consider the binary expansion of the counting function of irreducible monic polynomials of degree a given power of 2 over the field $$\mathbb{F}_{2}$$. Finally, we relate the problem of the irrationality of Euler constant with the binary expansion of the sum of the divisor function.

The Banach-Saks index of intersection

JPRM-Vol. 1 (2012), Issue 1, pp. 22 – 27 Open Access Full-Text PDF
Novikova A. I
Abstract: In this paper we estimate Banach-Saks index of intersection of two spaces with symmetric bases from below by indices of these spaces. We also show on example of Orlicz spaces that we can’t estimate Banach-Saks index of intersection in the same way from above.

Stability estimate for the multidimensional elliptic obstacle problem with respect to the obstacle function

JPRM-Vol. 1 (2012), Issue 1, pp. 12 – 21 Open Access Full-Text PDF
Abstract: The stability estimate of the energy integral established by Danelia, Dochviri and Shashiashvili [1] for the solution of the multidimensional obstacle problem in case of the Laplace operator is generalized to the case of arbitrary linear second order self-adjoint elliptic operator. This estimate asserts that if two obstacle functions are close in the $$L^{∞}$$-norm, then the gradients of the solutions of the corresponding obstacle problem are close in the weighted $$L^{2}$$ -norm.

Some more remarks on grothendieck-lidskii trace formulas

JPRM-Vol. 1 (2012), Issue 1, pp. 05 – 11 Open Access Full-Text PDF
Oleg Reinov
Abstract: Let $$r ∈ (0, 1]$$, $$1 ≤ p ≤ 2$$, $$u ∈ X^∗⊗X$$ and $$u$$ admits a representation $$u=\sum_{i}\lambda_{i}x_{i}^{‘}⊗ x_{i}$$ with $$(λ_i) ∈ l_r$$ bounded and $$(x_{i} ∈ l^{w}_{p’} (X)$$. If $$1/r + 1/2 − 1/p = 1$$ then the system $$\mu_{k}$$ of all eigenvalues of the corresponding operator $$\widetilde{u}$$ (written according to their algebraic multiplicities) is absolutely summable and trace $$u=\sum_{k}\mu_{k}$$. One of the main aim of these notes is not only to give a proof of the theorem but also to show that it could be obtained by A. Grothendieck in 1955.

A note on Stirling’s formula for the gamma function

JPRM-Vol. 1 (2012), Issue 1, pp. 01 – 04 Open Access Full-Text PDF
Dorin Ervin Dutkay, Constantin P. Niculescu, Florin Popovici
Abstract: We present a new short proof of Stirling’s formula for the Gamma function. Our approach is based on the Gauss product formula and on a remark concerning the existence of horizontal asymptotes.