Explicit Sierpiński and Riesel repunits and repdigits, and the bounded-factor problem

This paper develops explicit infinite families of Sierpiński and Riesel repunits and repdigits, together with rigorous reductions for the bounded-factor problem for decimal repunits. A Sierpiński number is a positive odd integer K for which K·2ⁿ + 1 is composite for every positive integer n; the Riesel definition uses K·2ⁿ − 1. The constructions use finite covering congruences whose periodicity certifies infinitely many coefficients with prescribed repeated-digit representations. Writing Rₜ(b) for the base-b repunit of length t, the principal families include: Sierpiński repunits R₍₂₉₃₄₅₊₆₉₈₈₈ⱼ₎(53). Riesel repunits R₍₇₇₇₉₅₊₂₀₄₆₀₀ⱼ₎(94) and R₍₉₂₁₉₅₊₂₀₄₆₀₀ⱼ₎(94). Sierpiński repdigits 17R₍₆₁₃₊₉₃₆ⱼ₎(40). Riesel repdigits 29R₍₆₀₇₉₊₈₂₀₈ⱼ₎(39). Each assertion holds for every nonnegative integer j. Additional constructions give Sierpiński repunits in base 66 and Riesel repunits in base 603. Together, the principal certificates establish upper bounds of 53, 40, 94, and 39 for the least bases admitting Sierpiński repunits, Sierpiński repdigits, Riesel repunits, and Riesel repdigits, respectively. The manuscript compares these with the corresponding published bounds of 147, 87, 16,518,444,216,571, and 180. These are improved constructive upper bounds, not determinations of the minimum possible bases. Complete finite enumeration classifies successful length phases within the specified covering-prime dictionaries. Polynomial congruences extend the constructions to further base families. A uniform obstruction shows that no finite prime dictionary can cover either exponential sequence associated with a binary repunit. This excludes that certification mechanism; it does not establish that every binary repunit fails the Sierpiński or Riesel definition. The second part studies whether infinitely many decimal repunits have at most a fixed number of prime factors, counted with multiplicity. It establishes an equivalence between the existential bounded-factor problem over arbitrary indices and over prime indices, derives multiplicative-order and valuation restrictions, strengthens a growing upper bound on total factor count using an established valuation theorem, and formulates exact finite-block factor-budget and endpoint-moment certificates. The paper also examines auxiliary-polynomial approaches. A consequence of the moving Subspace Theorem controls common factors for auxiliary polynomials of bounded degree and subexponential coefficient size. A pigeonhole construction shows why unrestricted degree growth invalidates this conclusion, even when the permitted degree grows arbitrarily slowly. Lattice reduction makes a substantial class of these counterexamples constructible. A separate elementary estimate permits growing binomial degree for almost all prime indices while retaining prime-power multiplicities. Exact rank-sieve identities, spectral calculations, relative-entropy thresholds, and structural countermodels identify limitations of several proposed transfers. The manuscript documents reproducibility through modular certificate replay, exhaustive phase witnesses, and exact arithmetic checks. The explicit covering families are proved; infinitude of decimal repunits with a fixed bounded total factor count remains open.