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Collatz Conjecture Proof Essay Using Binary System
Mathematical Essay to Prove the Collatz conjecture (CC) in infinite values using the binary numeral system.According to CC's For any number N from 1 to infinite,
If N is an odd number then do "3N+1".
If N is an even number then do "N/2" and repeat it as long as the obtained value is an even number.
Then Start all over with the newly obtained value until it goes down to 1
It has been known that, once the value 1 is reached, the said rules trigger an infinite loop of 4=>2=>1.
CC has been proven to be true so far for any number up to 2^48 using brute force method. However the brute force method cannot eliminate the theoretical existence of an "infinite" number that might disprove the CC..My attempt from this essay is to use a different method which is based on mathematical formulas that may be able to prove the CC in infinite infinite values, infinite combinations and infinite CC process run. Through the lenses of the binary numeral system, many facts were discovered such
- Expansion of Each Transitional CC Term At Its Head Is Bound
- Shrinkage of Each Transitional CC Term At Its Tail Is Unbound and Defined by Formula “2^n”
- Intrinsic Interchange Behavior of Binary Strings “01” vs. “11”
- Pro-CC Behavior Pattern of Chain of Binary Strings “01” at Tail of CC Unbound Digit Shrinkage Outruns Bound Digit Expansion
- Hidden Mathematical Correlation Between “3N” Operation and Fast Shrinkage Triggers
- Pro-CC Behavior Pattern of Uniform Binary Strings of “0”: Acceleration of Digit Shrinkage and Stopping/Slowdown of Formation of Anti-CC Binary Strings
- Pro-CC Behavior Pattern of Uniform Binary Strings of “1” at Tail of CC Self-Dissolution vs.
- Anti-CC Behavior Pattern of Uniform Binary Strings of “1” at Tail of Automatic Digit Expansion of CC Term, but this Anti-CC Behavior Pattern are overcome by its own pro-CC Behavior Pattern,
If N is an odd number then do "3N+1".
If N is an even number then do "N/2" and repeat it as long as the obtained value is an even number.
Then Start all over with the newly obtained value until it goes down to 1
It has been known that, once the value 1 is reached, the said rules trigger an infinite loop of 4=>2=>1.
CC has been proven to be true so far for any number up to 2^48 using brute force method. However the brute force method cannot eliminate the theoretical existence of an "infinite" number that might disprove the CC..My attempt from this essay is to use a different method which is based on mathematical formulas that may be able to prove the CC in infinite infinite values, infinite combinations and infinite CC process run. Through the lenses of the binary numeral system, many facts were discovered such
- Expansion of Each Transitional CC Term At Its Head Is Bound
- Shrinkage of Each Transitional CC Term At Its Tail Is Unbound and Defined by Formula “2^n”
- Intrinsic Interchange Behavior of Binary Strings “01” vs. “11”
- Pro-CC Behavior Pattern of Chain of Binary Strings “01” at Tail of CC Unbound Digit Shrinkage Outruns Bound Digit Expansion
- Hidden Mathematical Correlation Between “3N” Operation and Fast Shrinkage Triggers
- Pro-CC Behavior Pattern of Uniform Binary Strings of “0”: Acceleration of Digit Shrinkage and Stopping/Slowdown of Formation of Anti-CC Binary Strings
- Pro-CC Behavior Pattern of Uniform Binary Strings of “1” at Tail of CC Self-Dissolution vs.
- Anti-CC Behavior Pattern of Uniform Binary Strings of “1” at Tail of Automatic Digit Expansion of CC Term, but this Anti-CC Behavior Pattern are overcome by its own pro-CC Behavior Pattern,
103 pages, Kindle Edition
Published October 20, 2021
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