Papers by Vicente Padilla
arXiv: General Mathematics, 2019
In this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the sam... more In this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the same pattern: 1) Upward trajectory. Odd and even numbers alternate until the cycle reaches an upper bound 2) Downward trajectory. Two or more consecutive even numbers follow until it reaches another odd number. At this point, it's the beginning of the following cycle. Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle. Finally, we prove that both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a...
n this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the same... more n this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the same pattern: 1) Upward trajectory. Odd and even numbers alternate until the cycle reaches an upper bound 2) Downward trajectory. Two or more consecutive even numbers follow until it reaches another odd number. At this point, it's the beginning of the following cycle.
Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle.
Finally, we prove that both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.12. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.
In this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the sam... more In this paper; we prove that all sequences can be broken up in cycles. Each cycle follows the same pattern: 1) Upward trajectory. Odd and even numbers alternate until the cycle reaches an upper bound 2) Downward trajectory. Two or more consecutive even numbers follow until it reaches another odd number. At this point, it's the beginning of the following cycle.
Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle.
Finally, we prove that, if the number of cycles is large enough, then the initial number of both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.11. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.
We proved that all sequences can be broken up in cycles. We build two sequences. The first one st... more We proved that all sequences can be broken up in cycles. We build two sequences. The first one starts from any odd number and moves forward. The second sequence is built backwards, in other words, starting from1 up. Both sequences follow parallel paths. In other words, they have the same pattern of upward and downward steps. Finally, we prove that if the number of cycles is large enough, they both must be the same. Since the latter converges to 1, so must the former.
This paper establishes the path followed by any odd number in its first cycle of the sequence. It... more This paper establishes the path followed by any odd number in its first cycle of the sequence. It is also shown which numbers converge to 1 in one single cycle.
Collatz - Syrcuse Conjecture. This paper attempts to solve the problem by finding an equivalent s... more Collatz - Syrcuse Conjecture. This paper attempts to solve the problem by finding an equivalent sequence - Master Sequences - to any given sequence. The paper proves that ALL Master Sequences converge to 1.
Collatz - Syrcuse Conjecture. This paper attempts to solve the problem by finding an equivalent s... more Collatz - Syrcuse Conjecture. This paper attempts to solve the problem by finding an equivalent sequence - Master Sequences - to any given sequence. The paper proves that ALL Master Sequences converge to 1.
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Papers by Vicente Padilla
Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle.
Finally, we prove that both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.12. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.
Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle.
Finally, we prove that, if the number of cycles is large enough, then the initial number of both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.11. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.
Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle.
Finally, we prove that both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.12. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.
Any sequence is evidently made of many consecutive cycles. In order to prove the conjecture, we build two sequences. The first sequence starts from any odd number. The sequence unfolds one cycle after the another. After each cycle, we build a second sequence that is built following a parallel path to the previous one, but adapting the last cycle to ensure it converges down to 1. In other words, their cycles have the same pattern of upward and downward steps except for the very last cycle.
Finally, we prove that, if the number of cycles is large enough, then the initial number of both sequences must be the same. Since the latter sequence converges to 1, so must the former. This document is a revision of the previous rev.11. There are significant changes that were made in order to make it easier to follow. The foundations of the demonstration remain the same.