Hercher 2024 sum squarefree integers power two

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Christian Hercher, On the Sum of Squarefree Integers and a Power of Two. arXiv:2411.01964 (2024). The arXiv record (https://arxiv.org/abs/2411.01964, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Erdos conjectured that every odd n > 1 is the sum of a squarefree number and a power of two (Problem #11); previous numerical checks reached 10^7 (Odlyzko) and 1.4 * 10^9 (McCranie). Hercher extends the verification to all odd n < 2^50, more than 8 * 10^5 times further, using a highly parallel GPU algorithm that sieves squarefree numbers and tests representations. Theorem 5 records the outcome: for every odd 1 < n < 2^50 there is a squarefree s and an exponent 1 <= k <= 13 with n = s + 2^k, so only small powers of two are ever needed. Section 2 develops heuristics via inclusion-exclusion over the events A_k that n - 2^k is squarefree, computing a priori probabilities c_l that one of n - 2^1, ..., n - 2^l is squarefree (Table 1: c_1 = 0.8106, c_6 = 0.999999) and, extending to 7 <= l <= 20, an expected smallest needed exponent of about 1.2159. Table 3 lists the smallest odd n for which n - 2^1, ..., n - 2^m are all non-squarefree, the record-setting cases within the search range. The paper is purely computational and heuristic evidence for the conjecture; it also recalls the Granville-Soundararajan implication that the conjecture would give infinitely many non-Wieferich primes.

Source: https://arxiv.org/abs/2411.01964.

Bears on. #11

Results to transcribe.

• Theorem 5: For every odd 1 < n < 2^50 there exist a squarefree positive integer s and an integer 1 <= k <= 13 with n = s + 2^k.
• Table 1: Heuristic a priori probabilities c_l that one of n - 2^1, ..., n - 2^l is squarefree: c_1 = 0.810569..., c_2 = 0.975870..., up to c_6 = 0.999999...
• Expected exponent: The heuristic expected value of the smallest needed exponent k is about 1.215854...
• Table 3: Smallest odd n for which n - 2^1, ..., n - 2^m are all non-squarefree, for each m in the searched range.
• Algorithm (Section 3): Massively parallel GPU sieve of squarefree numbers combined with per-n representation testing, enabling the 2^50 range.