mathematical challenge, number theory, A005245, integer complexity, complexity theory, exponentiation, mathematics is inconsistent, arithmetic is inconsistent, inconsistency

Any comments are welcome: Karlis.Podnieks@lu.lv

**Mathematical Challenge**

**By Karlis Podnieks**

University of Latvia

Consider representing of natural numbers by using 1, +, * and brackets. One can prove easily that the best way of representing the powers of 3 is as follows:
All the other variants contain more than 3n 1's.
It is the best way at least for n≤39 – as verified by Janis Iraids. More about the context – see A005245 in the The On-Line Encyclopedia of Integer Sequences. February 26, 2011 Added later: H. Altman, J.
Zelinsky. J. Iraids, K. Balodis, J. Čerņenoks, M.
Opmanis, R. Opmanis, K. Podnieks. |

More about the
context – see: February 26, 2011 |

According to Gödel's First Incompleteness Theorem, any formal theory of natural numbers is either inconsistent, or incomplete. The final step: prove that the second clause can be dropped. More about the context – see the end of Section 6.1 of my book about Goedel's Theorem. Possibly, the first real step towards a solution is announced in:
where the following is announced:
"From this fact follows, in particular, that the
February 26, 2011 |

mathematical challenge, number theory, A005245, integer complexity, complexity theory, exponentiation, mathematics is inconsistent, arithmetic is inconsistent, inconsistency