Journal of the Ramanujan Mathematical Society
Volume 40, Issue 4, December 2025 pp. 389–393.
The finite products of shifted primes and Moreira's Theorem
Authors:
Pintu Debnath
Author institution:Department of Mathematics, Basirhat College, Basirhat 743 412, North 24th Parganas, West Bengal, India.
Summary:
Let r ∈ N and N = ∪{r}{i = 1} C{i}. Do there exist x, y ∈ N and i ∈ {1, 2, ⋯ , r}
such that {x, y, xy, x + y} ⊆ C{i}? This is still an unanswered question asked
by N. Hindman. Joel Moreira in [J. Moreira, Monochromatic sums and products
in N, Annals of Mathematics 185 (2017) 1069–1090] established a partial
answer to this question and proved that for infinitely many x, y ∈ N, {x, xy,
x + y} ⊆ Ci for some i ∈ {1, 2, … , r}, which is called Moreira's Theorem.
Recently, H. Hindman and D. Strauss established a refinement of Moreira's
Theorem and proved that for infinitely many y, {x ∈ N : {x, xy, x + y} ⊆ Ci} is
a piecewise syndetic set. In this article, we will prove infinitely many y ∈
FP (P − 1) and FP (P + 1) such that {x ∈ N : {xy, x + f (y) : f ∈ F} ⊆ Ci} is
piecewise syndetic, where F is a finite subset of xZ [x]. P is the set of
prime numbers in N and FP (P − 1) is the set of all finite products of
distinct elements of P − 1.
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