About

The Collatz Conjecture, often referred to as the 3x + 1 problem, represents one of the most notoriously deceptive unsolved problems in mathematics. First proposed by Lothar Collatz in 1937, the premise relies on a strictly deterministic piecewise function applied sequentially to a positive integer. Despite its elementary arithmetic constraints, the resulting sequences - dubbed "hailstone numbers" due to their volatile ascending and descending trajectories - exhibit extreme pseudo-randomness.

This computational tool executes the algorithmic iterations in real-time, leveraging arbitrary-precision integers (BigInt) to calculate infinite-bound states without machine precision degradation. It strictly plots the state changes, quantifying the algorithmic total steps and identifying the maximum peak value before the sequence inevitably collapses into the trivial 4→2→1 loop. Failure to account for sequence volatility can cause integer overflow in standard systems; thus, algorithmic normalization techniques and logarithmic visual scaling are applied.

Formulas

The deterministic operation defines a sequence a_i. Given an arbitrary positive integer n, the subsequent term is calculated via the piecewise function f(n):

{

n2 if n ≡ 0 (mod 2)3n + 1 if n ≡ 1 (mod 2)

Where:
n = The current operational integer state.
(mod 2) = The modulo operation identifying parity (even or odd integers).

Reference Data

Search Limit (N <)	Starting Integer	Total Steps	Maximum Peak Value
10	9	19	52
100	97	118	9,232
1,000	871	178	190,996
10,000	6,171	261	3,256,576
100,000	77,031	350	60,004,654
1,000,000	837,799	524	1,164,136,254
10,000,000	8,400,511	685	112,056,231,184
100,000,000	63,728,127	949	853,235,930,064
1,000,000,000	670,617,279	986	43,363,554,499,392

Frequently Asked Questions

The standard conjecture applies strictly to positive integers. Evaluating negative integers using the same logic introduces distinct closed loops. For instance, initiating with -1 immediately enters a loop of -1, -2, -1. Initiating with -5 enters the cycle -5, -14, -7, -20, -10, -5. Mathematics currently identifies three distinct known cycles for negative integers.

This characteristic behavior dictates the term "hailstone numbers". Because odd numbers are multiplied by 3 and incremented by 1, the value experiences a severe upward spike. Subsequently, this operation guarantees the resulting integer is even, enforcing at least one immediate division by 2, causing an abrupt mathematical descent.

No absolute boundary has been discovered. The peak value scales disproportionately with the starting integer. For example, initiating the algorithm with 27 yields a peak integer of 9,232 (an escalation of roughly 342x), whereas starting with 28 peaks identically at 28.

In sequences mapping volatile starting states (such as 27 or 837,799), the maximum peak can eclipse the standard trajectory values by tens of thousands of factors. A standard linear graph compresses the baseline fluctuations to a negligible flatline. Implementing a base-10 logarithmic scale normalizes exponential variations, providing visual parity to low-magnitude state changes.