Establish unconditional infinitude for fixed-slope or higher-erasure cyclic constructions

Establish unconditional infinitude of admissible prime lengths for the fixed-slope 2 cyclic triple-node and general multi-slope constructions, or, more generally, extend the unconditional infinitude result to multi-slope constructions correcting more than three node erasures.

Background

The fixed-slope construction requires 2 to be primitive modulo an odd prime n. Infinitely many such primes are predicted by Artin’s primitive-root conjecture, but the paper does not have an unconditional proof for this fixed base. For triple-node correction, the paper circumvents this difficulty by selecting a slope from {2,-3,-15} and applying Heath-Brown’s theorem, thereby proving an unconditional infinite family.

The more general multi-slope construction correcting rho-node erasures still assumes that 2 is primitive modulo n. The paper explicitly notes that the unconditional infinitude argument for the three-slope triple-node case does not identify a single slope occurring infinitely often and does not extend its infinitude conclusion to rho>3.