# 157

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## Number

$157$ (**one hundred and fifty-seven**) is:

- The $37$th prime number

- The $12$th emirp after $13$, $17$, $31$, $37$, $71$, $73$, $79$, $97$, $107$, $113$, $149$

- The $1$st positive integer $n$ for which $\phi \left({2 n + 1}\right) < \phi \left({2 n}\right)$:
- $\phi \left({2 \times 157 + 1}\right) = 144, \phi \left({2 \times 157}\right) = 156$

- The largest odd positive integer that cannot be expressed as the sum of exactly $4$ non-zero square numbers all of which are coprime.

- The $3$rd balanced prime after $5, 53$:
- $157 = \dfrac {151 + 163} 2$

- The $8$th prime number after $53$, $71$, $103$, $107$, $109$, $149$, $151$ which cannot be expressed as the difference between a power of $2$ and a power of $3$.

## Also see

*Previous ... Next*: Prime Number*Previous ... Next*: Emirp*Previous ... Next*: Balanced Prime*Previous ... Next*: Primes not Difference between Powers of 2 and 3

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $157$