Proof of Collatz Conjecture Using Division Sequence

The purpose of this study is to prove the Collatz conjecture using a theorem proving system. First, the division sequence is defined as an alignment of the number of times division by 2 is performed in the Collatz operation. Then, the star conversion is defined, which is a mapping from a specific division sequence to a division sequence. Here it is important to map to some division sequence, not which division sequence. The important point is that the finite length of the division sequence does not change before and after the star conversion. In theorem proving system, we considered two parallel methods: main-proof is a claim to a computer proposition that has the same meaning as the Collatz conjecture. Theorem proving support system “Idris” was used. Moreover, we sub-proved that the 12 “extended star conversion” are closed to the “Collatz operation”. Egison’s computer algebra system is used for proof. The results of the two methods achieved the goal of proving the Collatz conjecture using a theorem proving system.

Characteristics of the Cretaceous Magmatism in Huanghua Depression and Their Relationships with Hydrocarbon Enrichment

The Huanghua depression located in the hinterland of the Bohai Bay Basin in eastern China is a typical area for the research of multistage magmatic activities with hydrocarbon enrichment,many high-yield wells related to igneous rocks were discovered within the Cretaceous strata in recent years.However,the spatial and temporal distribution of Cretaceous igneous rocks remains unclear,and the relationships among magmatic sequence,lithology,and hydrocarbon enrichment have been poorly studied.In order to solve these problems,core observation,logging analysis,major element analysis,zircon U-Pb chronology,oil-bearing grade statistics and reservoir spaces analysis were used to subdivide the magmatism cycles and to investigate the characteristics of igneous reservoirs.Our results show that the Mesozoic magmatism in Huanghua depression started in 140.1±1.4 Ma and could be divided into two stages including the Early Cretaceous stage and Late Cretaceous stage.The Early Cretaceous magmatism principally developed basic-intermediate rocks in the north zone,and could be subdivided into 3 cycles with their forming ages of 140,125–119,and 118–111 Ma,respectively.By contrast,the late stage mainly developed intermediate-acid rocks centralized in the south zone and formed at 75–70 Ma.The GR and SP curves are good indicators for the contrast of different lithologies,lithofacies and for magmatic sequences division.Intensive magmatism may have an advantage to form reservoirs,since basalt in cycle III in the Early Cretaceous and dacite porphyrite in the Late Cretaceous have great exploration potential.Lithology and tectonic fractures have an important influence on the formation of reservoir spaces and hydrocarbon enrichment.The characteristics of Cretaceous magmatism and igneous reservoirs in Huanghua depression and adjacent areas are summarized,providing important information for igneous reservoirs research and oil-gas exploration in the Cretaceous in related areas.