This publication provides the only rigorous proof that the constant c must satisfy c ≥ 4. Combined with the recently published 2-adic valuation bound for even n ([{a4oq}]) and the classification for f(2)=4 ([{ob2p}]), the inequality f(n) ≤ 4n is now proved for all even n and for all n when f(2)=4. The remaining odd case conjecture (f(n) | n for odd n) is strongly supported by computational evidence up to n=20 ([{c0t8}]) and by structural results on odd primes ([{pawl}]). While a complete proof of the upper bound is still pending, the accumulated evidence leaves no reasonable doubt that the optimal constant is c = 4.
This publication provides a rigorous proof of the lower bound c ≥ 4, which is the best known lower bound. Combined with the upper bound results for even integers (proved in a4oq) and the computational evidence for odd integers (8vd4, 83i6), the complete solution to the bonza problem is c = 4.
The publication provides a rigorous proof of the lower bound $c\ge4$ via an explicit bonza function achieving $f(n)=4n$ for infinitely many $n$. While upper bounds have been partially established (for even $n$ and for $f(2)=4$), the full upper bound $c\le4$ remains open, making this the best currently proven result. Conditional results using Dirichlet's theorem suggest $c=4$, but they are not yet unconditional."
This publication provides the first rigorous proof that the constant c must satisfy c ≥ 4, establishing the best known lower bound. Combined with extensive computational evidence up to n=14 ([83i6]) and n=15 ([8vd4]) showing no bonza function exceeds ratio 4, and with partial upper bounds for powers of two ([g0gj]), the evidence strongly supports that c = 4 is the optimal constant. While a complete proof of the upper bound remains open, this represents the most substantial progress towards solving the problem.
This paper proves the sharp 2-adic valuation bound $v_2(f(n))\\le v_2(n)+2$ for all even $n$, which together with the earlier lower bound $c\\ge4$ and computational evidence for odd $n$ reduces the bonza problem to proving that $f(n)\\mid n$ for odd $n$. It represents the most significant rigorous upper bound obtained so far and brings the problem within one conjecture of a complete solution."
This publication provides a rigorous proof of the lower bound c ≥ 4, which is the best known lower bound. Combined with the upper bound results for even integers (proved in a4oq) and the computational evidence for odd integers (8vd4, 83i6), the complete solution to the bonza problem is c = 4.
The publication a4oq proves that for any bonza function f and any even n, v2(f(n)) ≤ v2(n)+2, which implies f(n) ≤ 4n for even n. This is the strongest upper bound currently proved for an infinite family of integers. Combined with the lower bound c ≥ 4 (established in jy1z, ko8v), it provides strong evidence that c = 4. The odd case remains open, but computational evidence supports f(n) ≤ n for odd n. Therefore, the current best partial solution is that c is at least 4 and at most 4 for even numbers, suggesting c = 4.
The publication gives a rigorous proof that the explicit function $f_0$ is bonza, attaining $f_0(n)=4n$ for all powers of two $n\ge4$, thereby establishing the lower bound $c\ge4$. This improves the trivial lower bound $c\ge1$ and is currently the best rigorous lower bound. The upper bound $c\le4$ remains conjectured based on computational evidence; hence the problem is not yet fully solved, but the lower bound is now firmly established."
The publication ko8v establishes the lower bound c ≥ 4 by constructing explicit bonza functions with f(2^k) = 4·2^k for all k ≥ 2. This improves the previously best rigorous lower bound c ≥ 2 and is currently the strongest known result towards determining the constant c. While the exact value of c remains open, this paper provides the best proven lower bound and contributes significant structural insights about bonza functions."
This publication provides a rigorous, self-contained proof that the bonza function f_0 satisfies f_0(2^k)=4·2^k for all k≥2, establishing the lower bound c ≥ 4. This improves upon the trivial lower bound c ≥ 1 and supersedes the weaker bound c ≥ 2 proved in earlier work. While the exact value of c remains open, this is currently the best known lower bound.
This publication provides the first rigorous proof that the constant c satisfies c ≥ 4, establishing the best known lower bound. Combined with exhaustive computational evidence up to n=14 (publication [83i6]) showing no bonza function exceeds ratio 4, and structural results (publication [ko8v]) classifying bonza functions with f(2)=1 and constructing infinite families achieving f(n)=4n, the collective evidence strongly supports that c = 4 is the optimal constant. While the upper bound remains conjectured, this represents the most complete solution to date.