# 37

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## Number

$37$ (**thirty-seven**) is:

- The $12$th prime number, after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$, $31$

- Every positive integer can be expressed as the sum of at most $37$ positive $5$th powers

- The $2$nd integer which has a reciprocal whose period is $3$:
- $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

- The $3$rd unique period prime after $3$, $11$: its period is $3$:
- $\dfrac 1 {37} = 0 \cdotp \dot 02 \dot 7$

- The $4$th centered hexagonal number after $1$, $7$, $19$:
- $37 = 1 + 6 + 12 + 18 = 4^3 - 3^3$

- The $4$th prime number of the form $n^2 + 1$ after $2$, $5$, $17$:
- $37 = 6^2 + 1$

- The $4$th emirp after $13$, $17$, $31$

- The $4$th prime $p$ after $11$, $23$, $29$ such that the Mersenne number $2^p - 1$ is composite

- The $6$th two-sided prime after $2$, $3$, $5$, $7$, $23$:
- $37$, $3$, $7$ are prime

- The $7$th integer $m$ such that $m! + 1$ (its factorial plus $1$) is prime:
- $0$, $1$, $2$, $3$, $11$, $27$, $37$

- The $8$th right-truncatable prime after $2$, $3$, $5$, $7$, $23$, $29$, $31$

- The $8$th left-truncatable prime after $2$, $3$, $5$, $7$, $13$, $17$, $23$

- The $9$th permutable prime after $2$, $3$, $5$, $7$, $11$, $13$, $17$, $31$

- The $11$th lucky number:
- $1$, $3$, $7$, $9$, $13$, $15$, $21$, $25$, $31$, $33$, $37$, $\ldots$

- The $19$th odd positive integer that cannot be expressed as the sum of exactly $4$ distinct non-zero square numbers all of which are coprime
- $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, $23$, $25$, $27$, $29$, $31$, $33$, $35$, $37$, $\ldots$

- The $23$rd positive integer after $2$, $3$, $4$, $7$, $8$, $\ldots$, $26$, $29$, $30$, $31$, $32$, $33$ which cannot be expressed as the sum of distinct pentagonal numbers

- The $27$th (strictly) positive integer after $1$, $2$, $3$, $\ldots$, $27$, $30$, $31$, $32$, $35$, $36$ which cannot be expressed as the sum of distinct primes of the form $6 n - 1$

- The $27$th integer $n$ such that $2^n$ contains no zero in its decimal representation:
- $2^{37} = 137 \, 438 \, 953 \, 472$

## Also see

- 37 is Second Number whose Period of Reciprocal is 3
- Numbers whose Cyclic Permutations of 3-Digit Multiples are Multiples
- Hilbert-Waring Theorem for $5$th Powers

*Previous ... Next*: Pythagorean Prime*Previous ... Next*: Sequence of Indices of Composite Mersenne Numbers

*Previous ... Next*: Prime Number*Previous ... Next*: Right-Truncatable Prime*Previous ... Next*: Emirp*Previous ... Next*: Permutable Prime

*Previous ... Next*: Numbers not Expressible as Sum of Distinct Pentagonal Numbers*Previous ... Next*: Lucky Number

*Previous ... Next*: Integers not Expressible as Sum of Distinct Primes of form 6n-1*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation

## Sources

- 1979: G.H. Hardy and E.M. Wright:
*An Introduction to the Theory of Numbers*(5th ed.) ... (previous) ... (next): $\text I$: The Series of Primes: $1.2$ Prime numbers - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $37$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $37$

Categories:

- Unique Period Primes/Examples
- Centered Hexagonal Numbers/Examples
- Left-Truncatable Primes/Examples
- Two-Sided Primes/Examples
- Prime Numbers/Examples
- Right-Truncatable Primes/Examples
- Emirps/Examples
- Permutable Primes/Examples
- Lucky Numbers/Examples
- Integers not Expressible as Sum of Distinct Primes of form 6n-1/Examples
- Specific Numbers
- 37