# 703

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## Number

$703$ (seven hundred and three) is:

$19 \times 37$

The second of the $3$rd pair of triangular numbers whose sum and difference are also both triangular:
$378 = T_{27}$, $703 = T_{37}$, $378 + 703 = T_{46}$, $703 - 378 = T_{25}$

The $5$th Fermat pseudoprime to base $3$ after $91$, $121$, $286$, $671$:
$3^{703} \equiv 3 \pmod {703}$

The $7$th Kaprekar number after $1$, $9$, $45$, $55$, $99$, $297$:
$703^2 = 494 \, 209 \to 494 + 209 = 703$

The $19$th hexagonal number after $1$, $6$, $15$, $28$, $45$, $66$, $91$, $\ldots$, $378$, $435$, $496$, $561$, $630$:
$703 = \ds \sum_{k \mathop = 1}^{19} \paren {4 k - 3} = 19 \paren {2 \times 19 - 1}$

The $37$th triangular number after $1$, $3$, $6$, $10$, $15$, $\ldots$, $325$, $351$, $378$, $406$, $435$, $465$, $496$, $528$, $561$, $595$, $630$, $666$:
$703 = \ds \sum_{k \mathop = 1}^{37} k = \dfrac {37 \times \paren {37 + 1} } 2$