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How to Find the Smallest Base for a 135-Digit Polynomial Value

Camila Milosovich Sep 30, 2026 869 views

Question details

Calculate the smallest integer base that converts a ten-character Unicode string into a value with at least 135 decimal digits.

How to Find the Smallest Base for a 135-Digit Polynomial Value

Product: Programming / Math
Device & OS: not provided
Scenario: Evaluating a polynomial based on string characters to reach a large numerical threshold (10^134).
Observed behavior: The goal is to determine the exact minimum base using Horner's method, binary search, and arbitrary-precision integer arithmetic to prevent floating-point inaccuracies.

Before you start

Ensure you have a programming environment like Python installed, as standard spreadsheet software cannot handle 135-digit arbitrary-precision arithmetic natively.

Solution 1Recommended

Use Horner's Method and Binary Search in Python

Write a Python script using built-in arbitrary-precision integers to efficiently evaluate the polynomial and search for the minimum base.

Because the target is a 135-digit number, the minimum required value is 10^134. Since the polynomial value strictly increases as the base increases (monotonicity), binary search is the most efficient way to pinpoint the exact smallest base.

Convert characters to Unicode coefficients

Extract the numerical value of each character in the 10-character string using the built-in `ord(character)` function.

Implement Horner's method

Write a function to evaluate the polynomial iteratively. Using the formula `value = value * base + ord(character)`, iterate through the string characters. This requires fewer multiplications than calculating powers directly.

Determine lower and upper bounds

Set your lower bound to 2. To find the upper bound, start with a small base and repeatedly increase it (e.g., doubling it) until the evaluated polynomial value meets or exceeds the target `10**134`.

Execute the binary search

Write a loop that calculates the midpoint between the lower and upper bounds. If the polynomial evaluated at the midpoint base is greater than or equal to `10**134`, lower the upper bound. Otherwise, raise the lower bound until the exact minimum base is found.

Use Horner's Method and Binary Search in Python

Arbitrary-Precision Arithmetic Required: You must use purely integer arithmetic for this computation. Floating-point arithmetic limits precision to 53 bits (in standard double precision) and will cause severe data loss long before reaching a 135-digit number.

Document Algorithms and Code Using WPS Writer

While spreadsheets cannot natively process 135-digit arbitrary-precision numbers, you can use WPS Writer to professionally document your algorithms, format Python code snippets, and display complex polynomial math.

1. Create a new document: Open WPS Office, select 'Writer', and click 'Blank Document' to start your technical documentation.
2. Insert math formulas: Navigate to the 'Insert' tab and click 'Equation' to typeset your base polynomial calculation clearly.
3. Format your code: Paste your Python script into the document, select the text, and use a monospaced font like Courier New from the 'Home' tab for readability.
4. Save and export: Click 'Menu' > 'Export to PDF' to create a clean, shareable document of your mathematical solution.

Rich text formatting for clean and readable code blocksBuilt-in Equation Editor for rendering Horner's method math formulasSeamless format compatibility with Microsoft Word documents (.docx)Free, lightweight, and perfect for technical documentation

microsoft office alternative - wps office

Frequently Asked Questions

Why is 10^134 used as the target for a 135-digit number?

In base-10, the smallest possible number with N digits is 10^(N-1). Therefore, the absolute smallest value that takes up 135 decimal digits is a 1 followed by 134 zeros, which is exactly 10^134.

What makes Horner's method better than standard polynomial evaluation?

Horner's method factors out the base at each step, restructuring the polynomial as `(...((a₀ × b + a₁) × b + a₂) ... × b + aₙ)`. This reduces the computational complexity from O(n^2) multiplications to O(n), which is significantly faster for computer algorithms.

Can I use Excel or WPS Spreadsheet to calculate this base?

Standard spreadsheet applications limit number precision to 15 significant digits (IEEE 754 standard). Because calculating a 135-digit polynomial requires exact arbitrary precision, you must use a programming language like Python that supports infinitely large integers.

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