# Euler’s Totient Function

### Steven Hofmann

**Euler’s Totient Function***, 2013*

*Acrylic on Canvas*

*24 inches x 36 inches*

*$1700*

Math is referred to as the universal language. Mathematicians seek patterns and use them to formulate conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning provides predictions about nature.

Seventy-two is the sum of Euler’s Totient Function over the first fifteen integers. Seventy-two is also a pronic number. The sum of four consecutive primes is 72 (13+17+19+23) and six primes (5+7+11+13+17+19). Leonhard Euler introduced the function in 1760. The standard notation (n) is from Gauss’ 1801 treatise Disquisitiones Arithmeticae. Thus, it is usually called Euler’s phi function or the phi function.

In 1879 J. J. Sylvester coined the term totient for this function, so it is also referred to as the totient function, the Euler totient, or Euler’s totient. Jordan’s totient is a generalization of Euler’s.

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