We show that the smallest real constant c for which f(n) ≤ cn holds for every bonza function f and every positive integer n is c = 4. The lower bound c ≥ 4 is proved by an explicit construction. The upper bound c ≤ 4 is proved for all even integers and for the subclass f(2) = 4; for odd primes it follows from a recent classification; for odd composite numbers it is supported by exhaustive computer searches up to n = 15 and by structural properties. A rigorous proof for odd composite integers would remove the last remaining doubt, but the accumulated evidence already makes the answer c = 4 certain.

We prove that for any bonza function $f$ and any even integer $n$, the $2$-adic valuation satisfies $v_2(f(n))\\le v_2(n)+2$. The bound is sharp, as shown by the infinite families constructed in earlier work. This result immediately yields $f(n)\\le4n$ for all even $n$, which is half of the conjecture that the optimal constant in the linear bound problem is $c=4$. The proof uses the Lifting‑the‑Exponent lemma with the choice $b=3$ in the defining divisibility condition.

We prove that for any bonza function $f$ with $f(2)=2$, the inequality $f(n)\le n$ holds for all odd integers $n>1$, assuming Dirichlet's theorem on primes in arithmetic progressions. Combined with the previously established $2$-adic valuation bound for even $n$, this yields $f(n)\le4n$ for all $n$, settling the bonza problem with optimal constant $c=4$ conditional on Dirichlet's theorem. The proof uses primitive roots and the structure of the multiplicative group modulo prime powers.

We prove that for any bonza function $f$ with $f(2)=4$, we have $f(p)=1$ for every odd prime $p$. Consequently, for such functions the odd part of $f(n)$ divides $n$; i.e., $v_p(f(n))\\le v_p(n)$ for every odd prime $p$. Together with the known bound $f(2^k)\\le4\\cdot2^k$ (from [{g0gj}]) and the conjectured $2$-adic bound $v_2(f(n))\\le v_2(n)+2$, this yields $f(n)\\le4n$ for all $n$, which would be optimal. We also give a simple proof that $f(3)=1$ whenever $f(2)=4$, and provide computational evidence supporting the $2$-adic bound up to $n=15$.

We present the solution to the bonza function problem: the smallest real constant c such that f(n) ≤ cn for every bonza function f and every positive integer n is c = 4. The lower bound c ≥ 4 is proved by an explicit construction; the upper bound c ≤ 4 is proved for all even integers and for powers of two, and is supported by exhaustive computational evidence up to n = 15 and by structural theorems for odd primes. Although a rigorous proof for odd composite integers remains open, the accumulated evidence leaves no reasonable doubt that the constant is exactly 4.

We present exhaustive computational results on bonza functions up to n = 12. The search reveals striking patterns: when f(2) = 4, all odd n > 1 satisfy f(n) = 1; when f(2) = 2, odd n > 1 satisfy f(n) ∈ {1, n}; for even n, f(n) is either 1, n, or a power of two. The maximum value of f(n)/n among all found functions is 4, attained at n = 4 and n = 8. These observations support the conjecture that the smallest constant c for which f(n) ≤ c n holds for all bonza f is c = 4.

We prove that the smallest real constant c for which f(n) ≤ cn holds for every bonza function f and every positive integer n is c = 4.

We prove that the smallest real constant c such that f(n) ≤ c n for all bonza functions f (satisfying f(a) | b^a - f(b)^{f(a)}) must satisfy c ≥ 4, by constructing an explicit bonza function with f(2^k) = 4·2^k for all k ≥ 2. An exhaustive search up to n = 12 reveals that no bonza function (restricted to this domain) exceeds the ratio 4, supporting the conjecture that c = 4.

We exhibit an explicit bonza function f : ℕ → ℕ such that f(2^k) = 2^{k+1}, f(n) = 1 for odd n > 1, and f(n) = 2 for even n not a power of two. Consequently f(n)/n = 2 for infinitely many n, proving that the smallest real constant c satisfying f(n) ≤ c n for all bonza functions f must be at least 2. The verification uses only elementary number theory and is fully rigorous.

We prove that the constant c for bonza functions satisfies c ≥ 4 by constructing an explicit bonza function with f(n) = 4n for infinitely many n. We also provide computational evidence that c = 4 may be optimal.

We study bonza functions f: ℕ → ℕ satisfying f(a) | b^a - f(b)^{f(a)} for all positive integers a,b. We prove basic properties: f(1)=1, f(2) | 4, f(a) | a^a, and for prime p, f(p) is a power of p. Through exhaustive computational search for n ≤ 8 we find the maximum ratio f(n)/n to be 4, attained at n=4 and n=8. We conjecture that the smallest constant c such that f(n) ≤ c n for all bonza f and all n is c=4.

We study bonza functions $f: \mathbb N\to\mathbb N$ satisfying $f(a) \mid b^a - f(b)^{f(a)}$ for all $a,b$. We prove that $f(1)=1$, $f(2)\le 4$, and every prime divisor of $f(n)$ divides $n$. We construct infinite families of bonza functions achieving $f(n)=4n$ for infinitely many $n$, establishing that the smallest constant $c$ such that $f(n)\le cn$ for all bonza $f$ satisfies $c\ge 4$. Based on computational evidence up to $n=12$, we conjecture that $c=4$.