This project is a C++ implementation of my papers:

1. "Improved Algorithms for Integer Complexity"[1] for computing the integer complexity $f(n)$ of a single integer $n$;
2. "Integer Complexity: Algorithms Meet Horner's Rule"[2] for computing the upper bound constant of integer complexity, heuristic algorithms, as well as other results. For more context, see OEIS A005245.

[1] Qizheng He, Improved Algorithms for Integer Complexity, arXiv:2308.10301 [cs.DS], 2023. [2] Qizheng He, Integer Complexity: Algorithms Meet Horner's Rule, arXiv: [cs.DS], 2026.

Open Conjectures

1. $f(2^i)=2i$?
2. $f(2^i3^j5^k)=2i+3j+5k$ ($k\leq 5$)?
3. $f(2^i+1)=2i+1$? (except 3 and 9)
4. $f(p^i)=i\cdot f(p)$, for $p=109,433,163,487,2$?

Single-Target Algorithm

Complexity

The theoretical running time of the single-target algorithm is $O(n^{0.6154})$. For our applications (searching for counterexamples for various number-theoretical conjectures) it is usually much faster, because of practical pruning strategies.

Implementations

For integers within long long ($\sim 2^{63}$): see single64.cpp.

For larger integers within u128, see single.cpp. It used C-RHO and C-Quadratic-Sieve for factoring. It also supports printing the formulas.

Run compile.bat to compile.

Subtractions

The code also supports computing the integer complexity using the operators $+,-,\times,()$. See A091333.

Upper Bound

We designed a more efficient algorithm that can compute an upper bound for the integer complexity, which holds for a set of numbers with natural density 1. The result is correct with high probability.

upper_bound_slow.cpp: $O(n^3/w)$ per sample, using base $2^{n_1}3^{n_2}$.

upper_bound.cpp: $O(n^2)$ per sample, using base $2^{n_1}3^{n_2}$.

upper_bound_3D.cpp: $O(n^3)$ per sample, using base $2^{n_1}3^{n_2}5^{n_3}$.

upper_bound_slow_4D.cpp: using base $2^{n_1}3^{n_2}5^{n_3}7^{n_4}$.

upper_bound_slow_5D.cpp: using base $2^{n_1}3^{n_2}5^{n_3}7^{n_4}11^{n_5}$. Supports finding the optimal $(n_1,n_2,n_3,n_4,n_5)$.

Heuristic algorithm

See upper_bound_3D.cpp.

TODO

1. Use a better factorization library.
2. Use parallelization.
3. Improve the lower bound computation.
4. Need u256 and more memory for a significantly larger range.

License

This project is shared under the terms of the GNU General Public License.