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09S Primitive Roots: Revisited

09. PEN B6 Suppose that $m$ does not have a primitive root. Show that

$a{}^{{}^\frac{\varphi(m)}{2}} \equiv 1 \; (mod \; m)$

for every $a$ relatively prime to $m$.

Here is the official solution file: pen09.
You can also disscuss the problems here!

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09. Primitive Roots: Revisited

09. PEN B6 Suppose that $m$ does not have a primitive root. Show that

$a{}^{{}^\frac{\varphi(m)}{2}} \equiv 1 \; (mod \; m)$

for every $a$ relatively prime to $m$.

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08S An arithmetic partition

08. PEN O 35 ( Romania TST 1998 )

Let $n \ge 3$ be a prime number and $a_{1} < a_{2} < \cdots < a_{n}$ be integers.
Prove that $a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $\{0, 1, 2, \cdots \}$ into sets $A_{1},A_{2},\cdots,A_{n}$ such that

$a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n}$

where $x + A$ denotes the set $\{x + a \vert a \in A \}$.

Here is the official solution file: PEN08S
You can also disscuss the problems here!

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08. An arithmetic partition

08. PEN O 35 ( Romania TST 1998 )

Let $n \ge 3$ be a prime number and $a_{1} < a_{2} < \cdots < a_{n}$ be integers.
Prove that $a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $\{0, 1, 2, \cdots \}$ into sets $A_{1},A_{2},\cdots,A_{n}$ such that

$a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n}$

where $x + A$ denotes the set $\{x + a \vert a \in A \}$.

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S07 A combinatorial congruence

07. PEN D2 (Putnam 1991/B4) Suppose that $p$ is an odd prime. Prove that

$\sum_{j = 0}^{p}\binom{p}{j}\binom{p + j}{j}\equiv 2^{p} + 1 \pmod{p^{2}}.$

Here is the official solution file: PEN07sol
You can also disscuss the problems here!

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