Persistence Calculator

Steps to single digit

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About Persistence Calculator

A persistence calculator computing both multiplicative persistence (multiply digits until single digit) and additive persistence (sum digits until single digit). Shows step chains, record holders, and conjectures. Client-side.

Persistence Calculator Features

  • Multiplicative chain
  • Additive chain
  • Both types
  • Record holders
  • Conjecture
Persistence: how many times must you multiply (or add) digits to reach a single digit? Multiplicative: 679→378→168→48→32→6 (5 steps). The record: 277777788888899 has persistence 11. Sloane conjectures no number has multiplicative persistence > 11.

How to Use

Enter a number:

  • Multiplicative: Product of digits chain
  • Additive: Sum of digits chain
  • Steps: Count for each

Record Holders

Multiplicative persistence records: 0→0, 10→1, 25→2, 39→3, 77→4, 679→5, 6788→6, 68889→7, 2677889→8, 26888999→9, 3778888999→10, 277777788888899→11. No 12+ known!

The Conjecture

Sloane's conjecture: maximum multiplicative persistence is 11. Verified for all numbers up to 10^233. Despite massive computation, no number with persistence > 11 has been found. Still unproven!

Step-by-Step Instructions

  1. 1Enter number.
  2. 2Compute chains.
  3. 3Count steps.
  4. 4Compare types.
  5. 5Check records.

Persistence Calculator — Frequently Asked Questions

What's the difference between the two types?+

Multiplicative: multiply all digits (679 → 6×7×9=378). Additive: sum all digits (679 → 6+7+9=22). Multiplicative persistence is much harder to achieve high values — record is 11. Additive persistence can be arbitrarily high (just add more 9s).

Why is multiplicative persistence bounded?+

Multiplication shrinks numbers MUCH faster than addition. Multiplying d digits gives at most 9^d, which has ≈d·log(9)/log(10) ≈ 0.954d digits. So the number of digits decreases geometrically — very few steps suffice.

What is the digital root connection?+

Additive persistence counts steps; digital root is the final value. They're related but different. Digital root of n = 1+(n-1)%9 (instant formula), but additive persistence requires actually counting iterations. No shortcut formula for persistence exists.

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