Recreational Mathematics

Single-Letter Representations Generator

Retrieve verified identities and construct new representations beyond 11111

Created by Inder J. Taneja
Definition. The letter a may have any value from 1 to 9, while adjacent letters indicate decimal concatenation.
11 = aa/a   ·   111 = aaa/a   ·   1111 = aaaa/a   ·   …
In general, a block of n letters satisfies a…a/a = (10ⁿ−1)/9. Every stored expression uses only addition, subtraction, multiplication and division, and produces the same natural number for all nine values of a.
Running-type collection. Searches the author’s manually constructed running-type identities.

Results

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Enter a natural number or choose an interval.

About the generator

The verified collections remain the preferred source. When a requested number is not stored, the generator compares two independent construction methods: a pattern-aware engine inferred from the author’s patterned families, and a structural engine using repeated-letter repunits, powers of ten, multiplication, factorisation and balanced decimal decomposition. It retains the candidate using fewer a characters, then fewer operations and shorter notation. Every displayed identity is checked with exact integer arithmetic for every value a = 1,…,9.

Foundational works

The generator is based on the author’s following two works:

  1. Inder J. Taneja, Single Letter Representations of Natural Numbers, Palindromic Symmetries and Number Patterns, RGMIA Research Report Collection, 18 (2015), Art. 40, pp. 1–30. RGMIA PDF.
  2. Inder J. Taneja, Single Letter Fraction-Type Representations of Natural Numbers-I, RGMIA Research Report Collection, Art. 149, September 27, 2017, pp. 1–136. RGMIA PDF.

Further references

  1. Inder J. Taneja, Single Letter Representations of Natural Numbers from 1 to 11111, Zenodo, February 5, 2019. doi:10.5281/zenodo.2557025.
  2. Inder J. Taneja, Fraction-Type Single Letter Representations of Natural Numbers From 1 to 11111, Zenodo, February 4, 2019. doi:10.5281/zenodo.2556902.
  3. Inder J. Taneja, Single Letter Patterned Representations and Fibonacci Sequence, Zenodo, February 6, 2019. doi:10.5281/zenodo.2558522.
  4. Inder J. Taneja, Patterned Single Letter Representations of Natural Numbers, Zenodo, July 2, 2020, pp. 1–110. doi:10.5281/zenodo.3928507.

numbers-magic.com · inderjtaneja.wordpress.com · Mathematical Generators