user15182: I hear that each address has one private key.
Is it possible generate three public key address by only one private key?

user12034: Not directly, no. Each private key corresponds to exactly one public key. This public key can be represented in two ways that give different addresses: uncompressed and compressed. This gives you two addresses for a private key.
You could, however, use the private key as a seed to generate more private keys, much like Electrum's 128-bit seed. For example, if your seed were the 24-bit 6B8ABC in hex, use the following numbers as private keys

SHA256(6B8ABC00)
SHA256(6B8ABC01)
SHA256(6B8ABC02)

(note: do not use a 24-bit seed or private key, as someone could easily steal your bitcoins)

user53: What do you mean by "public key address" ?
From one private key, you can generate:

A pay-to-pubkey hash address that uses the uncompressed form of the public key.
Pay-to-script hash address, where the script is (1)
Pay-to-script hash address, with a  OP_CHECKSIG script
Pay-to-script hash addresses, paying to 1-of-1 OP_CHECKMULTISIG with uncompressed pubkey.

And all of the above could use the compressed form of the pubkey to get another 4 addresses.
So one private key can be represented by eight different addresses.

15182: <p>I hear that each address has one private key.
Is it possible generate three public key address by only one private key?</p>
 12034: <p>Not directly, no. Each private key corresponds to exactly one public key. This public key can be represented in two ways that give different addresses: uncompressed and compressed. This gives you two addresses for a private key.</p>

<p>You could, however, use the private key as a seed to generate more private keys, much like <a href="https://electrum.org/seed.html" rel="nofollow">Electrum's 128-bit seed</a>. For example, if your seed were the 24-bit 6B8ABC in hex, use the following numbers as private keys</p>

<ul>
<li>SHA256(6B8ABC00)</li>
<li>SHA256(6B8ABC01)</li>
<li>SHA256(6B8ABC02)</li>
</ul>

<p>(note: do not use a 24-bit seed or private key, as someone could easily steal your bitcoins)</p>
 53: <p>What do you mean by "public key address" ?</p>

<p>From one private key, you can generate:</p>

<ol>
<li>A pay-to-pubkey hash address that uses the uncompressed form of the public key.</li>
<li>Pay-to-script hash address, where the script is (1)</li>
<li>Pay-to-script hash address, with a  OP_CHECKSIG script</li>
<li>Pay-to-script hash addresses, paying to 1-of-1 OP_CHECKMULTISIG with uncompressed pubkey.</li>
</ol>

<p>And all of the above could use the compressed form of the pubkey to get another 4 addresses.</p>

<p>So one private key can be represented by eight different addresses.</p>
