This site presents some fascinating insights into Collatz sequences. To explore these findings, a specialized computer tool was developed, making it accessible to anyone interested in delving deeper into this deceptively simple yet elusive mathematical puzzle. If you're unfamiliar with the Collatz problem, this is your last chance to turn back before falling into a true mathematical rabbit hole.
First, let's explain the problem. Or if you prefer, just watch this explanatory video from Veritasium.
Any positive integer can generate a Collatz sequence, following simple rules to calculate the next number: if the number is even, it is halved (x/2); if it is odd, it is multiplied by 3 and incremented by 1 (3x + 1). For example, starting with 5, the sequence is: 16, 8, 4, 2, 1, 4, 2, 1, and so on, with the cycle 4 → 2 → 1 repeating indefinitely.
The Collatz conjecture states that all such sequences eventually converge to the 4 → 2 → 1 loop, but no mathematical proof exists to confirm this. To disprove it, one would need to identify a starting number that produces a sequence diverging from this loop, either by increasing indefinitely or by entering a different loop. Alternatively, proving the conjecture requires a rigorous mathematical demonstration.
Despite this simplicity, the resulting sequences exhibit no clear pattern or trend. Instead, numbers within a sequence appear to rise and fall randomly. The only consistent behavior observed across all tested sequences up to 2^68 is that they eventually reach the 4 → 2 → 1 loop. However, the reason for this convergence remains elusive.
The Collatz problem can be thought of as a vast mathematical fractal, with different regions exhibiting unique, intricate patterns. Our view into this fractal is limited, and we see only fragments of its structure. By examining and piecing these discrete views together, we hope to uncover hidden rules or patterns that might explain the behavior of Collatz sequences.
To keep things concise, the findings are categorized into Main and Alternate. If you’re only interested in the key points, simply read the Main Findings page. If still engaged, go for the Alternate Findings.
The web version of the tool supports numbers up to 1,000,000, providing plenty of data for your observations. For numbers of any size, the full version is available for Windows or Mac upon request. The tool consists of four sections: Graph, Grid, Binary, and Export. A Manual for each section is provided to help you understand how to use them.
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