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Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture

Author

Listed:
  • Riccardo Gatti

    (National Laboratory of Molecular Biology and Stem Cell Engineering, Istituto Nazionale di Biostrutture e Biosistemi (INBB) c/o Eldor Lab, Via di Corticella 183, 40128 Bologna, Italy)

Abstract

Let S = s 1 , … , s n be a finite sequence of integers. Then, S is a Gilbreath sequence of length n , S ∈ G n , iff s 1 is even or odd and s 2 , … , s n are, respectively, odd or even and min K s 1 , … , s m ≤ s m + 1 ≤ max K s 1 , … , s m , ∀ m ∈ 1 , n . This, applied to the order sequence of prime number P , defines Gilbreath polynomials and two integer sequences, A347924 and A347925, which are used to prove that Gilbreath conjecture G C is implied by p n − 2 n − 1 ⩽ P n − 1 1 , where P n − 1 1 is the n − 1 -th Gilbreath polynomial at 1.

Suggested Citation

  • Riccardo Gatti, 2023. "Gilbreath Equation, Gilbreath Polynomials, and Upper and Lower Bounds for Gilbreath Conjecture," Mathematics, MDPI, vol. 11(18), pages 1-7, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:4006-:d:1244394
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