 Show that, If p is an odd prime, then there exist integers a, b, k such that a²+b+1=kp and 0<k<p.

To prove that for an odd prime \( p \), there exist integers \( a \), \( b \), and \( k \) such that

\[
a^2 + b + 1 = kp
\]
and
\[
0 < k < p,
\]

we can proceed with the following steps:

### Step 1: Consider quadratic residues modulo \( p \)
A key insight comes from the study of quadratic residues modulo \( p \). For any integer \( a \), the values \( a^2 \mod p \) will give the quadratic residues mod \( p \). Since \( p \) is odd, there are exactly \( \frac{p+1}{2} \) distinct quadratic residues modulo \( p \), and the remaining \( \frac{p-1}{2} \) numbers are non-residues.

### Step 2: Equation modulo \( p \)
We are looking for integers \( a \), \( b \), and \( k \) such that

\[
a^2 + b + 1 \equiv 0 \pmod{p}.
\]

This is equivalent to

\[
a^2 + b \equiv -1 \pmod{p}.
\]

For each value of \( a^2 \mod p \), we want to check if we can find \( b \) such that

\[
b \equiv -1 - a^2 \pmod{p}.
\]

### Step 3: Choice of \( k \)
Once we have \( a \) and \( b \) such that \( a^2 + b + 1 = kp \), we can solve for \( k \) by dividing both sides of the equation by \( p \). Since \( a^2 + b + 1 \equiv 0 \pmod{p} \), we know that the left-hand side is divisible by \( p \), and hence

\[
k = \frac{a^2 + b + 1}{p}
\]

will be an integer. Moreover, \( k \) satisfies \( 0 < k < p \) because \( a^2 + b + 1 \) is less than \( p^2 \) but greater than 0, thus ensuring that \( k \) remains in the desired range.

### Step 4: Existence of \( a \) and \( b \)
We need to show that for any \( p \), we can always find suitable \( a \) and \( b \). Since the set of quadratic residues and the integers modulo \( p \) cover all values modulo \( p \), and because \( -1 \) is always a

Show that, If p is an odd prime, then there exist integers a, b, k such that a² + b + 1 = kp and 0 < k < p.

This problem shows how to prove the existence of integers a, b, and k such that a² + b + 1 = kp and 0 < k < p, where p is an odd prime. The solution involves modular arithmetic and the concept of quadratic residues. Suitable for undergraduate students studying number theory.

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