A note on my 2012 paper "Patterns related to the Smarandache circular sequence primality problem"

A note on my 2012 paper “Patterns related to the Smarandache circular sequence primality problem” Marco Ripà October 2025 Abstract This short note restates, in a compact and rigorous way, one of the main results first presented in Ripà (2012), Patterns related to the Smarandache circular sequence primality problem, published in Notes on Number Theory and Discrete Mathematics. Relying on the circular permutations of the concatenated sequence S ( r ) = 123 . . . r (see OEIS A007908 for the base concatenation and A001292 for its circular permutations), that work examined the distribution of prime numbers within those arrangements, with particular attention to the positions of terms divisible by fixed primes. By constructing an ad hoc modular sieve that filtered out all multiples of smaller primes, the paper revealed, through explicit computation and graphical representation, the presence of perfectly periodic modular “tiles” describing the divisibility patterns arising from such rotations. Here, the same phenomenon is reformulated in modern terms, showing how it follows from the modular properties of decimal concatenation, repunits, and multiplicative orders. MSC2020: 11Y11 (Primary); 11A07, 11A63, 11B83 (Secondary). Keywords: Modular Tessellations, Repunits, Prime Divisibility Patterns. 1 Introduction As in [1], we work with the Smarandache consecutive sequence [3] S(r) := 1 2 3 . . . r (r∈Z+), obtained by juxtaposing the first r positive integers in base 10. Following Definition 1.1 of [ 1 ], we consider the set M ( r )of circular permutations of integer blocks of S ( r )(see [ 2 ]): these are obtained by cyclically reordering the blocks 1 , 2 , . . . , r while leaving each block internally unchanged (no digit-level permutations). Notation change for clarity. In [ 1 ], the rightmost block of a given permutation of M ( r ) was indexed by the symbol p . To avoid conflict with the notation for prime numbers, in this note, we denote by b∈ { 1 , 2 , . . . , r} the index of the block that appears at the right end of the chosen permutation in M ( r ). For example, the permutation r_ 1 _ 2 _ 3 _..._ ( r− 1) corresponds to b=r−1. Thus, every element of M ( r )contains exactly the same digits as S ( r ), merely rearranged by circular block rotations. 1 Digit rotations inside a block. For any block X belonging to M ( r ), let ℓ ( X )denote its number of digits in base 10. For each integer k with 0 ≤k < ℓ ( X ), define Rk ( X )as the result of a left rotation by kdigits within X. In the residue ring Z (10ℓ(X)−1)Zthis operation satisfies Rk(X)≡X·10k(mod 10ℓ(X)−1).(1) Hence, rotating the digits of a block is arithmetically equivalent to multiplication by a power of 10 modulo 10ℓ(X)−1(see [6]). 2 Divisibility under all rotations Let pj denote the j -th prime, and assume pj∤ 10. From (1) we obtain the following facts. (Sufficiency). For any integer block Xoccurring in M(r)and any 0≤k < ℓ(X), pj|gcdX, 10ℓ(X)−1=⇒pj|Rk(X); (2) indeed, if pj|X and pj| (10 ℓ(X)− 1), then by (1) we have Rk ( X ) ≡ 0 ( mod pj )for all k . (Equivalence, under pj| (10 ℓ(X)− 1)). If pj| (10 ℓ(X)− 1), then for any block X of M ( r ), ∀k, pj|Rk(X)⇐⇒ pj|X⇐⇒ pj|gcdX, 10ℓ(X)−1.(3) The forward direction follows by taking k= 0; the reverse direction follows from (2). Hence, whenever pj| (10 ℓ(X)− 1) and pj|X , the same prime divides all rotations of X . Such primes pj correspond to full columns in the divisibility grids considered in [ 1 ]: for each fixed block index b such that pj|Xb , every rotation offset k is marked, meaning that the column is entirely filled. 3 Periodicity across circular permutations Fix r∈Z+ and consider S ( r ) = 1 2 3 . . . r , whose circular block permutations form the set M(r). Each element of M(r)is uniquely determined by the index bof the block that appears at the right end of the permutation, according to Definition 1.1 of [1]. Thus, varying bfrom 1to rproduces all possible circular permutations of S(r). For each admissible block index b and each integer k with 0 ≤k < ℓ ( X ), we define the Boolean function Mpj(b, k) :=          1if the rightmost block indexed by b, after a left rotation by kdigits, satisfies Rk(Xb)≡0 (mod pj), 0otherwise. Hence, Mpj ( b, k )takes the value 1precisely when the k -th digit rotation of the block Xb in the circular permutation of S ( r )indexed by b is divisible by the prime pj . Equivalently, it provides a binary map encoding all congruences Rk(Xb)≡0 (mod pj), 2 and therefore determines the complete two-dimensional divisibility pattern generated by the set of blocks of M(r)and their internal digit rotations. Here, νpj (10) denotes the multiplicative order of 10 modulo pj , and the following periodicities hold: (i) Along the internal-rotation axis k, the period is ℓ(Xb), since Rk+ℓ(Xb)(X) = Rk(X). (ii) Along the permutation index b , periodicity divides the multiplicative order νpj (10), because consecutive values of b change the relative decimal weight of each block by a power of 10, and these powers repeat modulo pjafter νpj(10) steps. (iii) Therefore, the grid of true values of Mpj forms a rectangular lattice of fundamental periods νpj(10), ℓ(Xb)in the (b, k)-plane. (iv) If pj| (10 ℓ(Xb)− 1), then by (3) each vertical line corresponding to a fixed b such that pj|Xbis entirely filled: every rotation of that block is divisible by pj. This two-dimensional periodic structure, first visualised in Figure 4 of [ 1 ], corresponds to the modular tessellations displayed (as particular cases) in Figures 2, 3, and 6–20 of the same paper, which exhibit perfect “arithmetical tiles” for the primes 7,11, and 13. The repetition of these tiles across the ( b, k )-plane explains the observed symmetry patterns for each prime modulus pj≥7. 4 Conclusion The tessellations discussed in Sections 2–3follow directly from the classical arithmetic of repunits and decimal rotations (see [ 6 ]). The original paper [ 1 ] first demonstrated these regularities by explicit construction and graphical representation (see also the derived OEIS sequences [ 4 , 5 ]), anticipating the lattice formulation presented here. Finally, for any integer base g > 2, the same reasoning carries over to radixg by replacing 10 with g throughout. References [1] Ripà, M. (2012). Patterns related to the Smarandache circular sequence primality problem. Notes on Number Theory and Discrete Mathematics, 18(1), 29–48. [2] OEIS Foundation Inc. (2025). A001292. The Online Encyclopedia of Integer Sequences. Available at: https://oeis.org/A001292. [3] OEIS Foundation Inc. (2025). A007908. The Online Encyclopedia of Integer Sequences. Available at: https://oeis.org/A007908. [4] OEIS Foundation Inc. (2025). A180346. The Online Encyclopedia of Integer Sequences. Available at: https://oeis.org/A180346. [5] OEIS Foundation Inc. (2025). A181373. The Online Encyclopedia of Integer Sequences. Available at: https://oeis.org/A181373. [6] Hardy, G. H. and Wright, E. M. (1979). An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford. 3