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[PHP] Trouble with PEAR Capatcha

Hello. I am trying to write a form with capatcha but run into trouble.
Below is what my error_log tells me. I am using LAMP on a Ubuntu Desktop.
(Laters version)

The link to the <img src="capatcha.php"> is broken or the error is in Line
42 that is like this:

file_put_contents(md5(session_id()) . '.png', $png);

[Thu Mar 31 16:28:39.294818 2016] [:error] [pid 10425] [client] PHP Warning:
file_put_contents(d41d8cd98f00b204e9800998ecf8427e.png): failed to open
stream: Permission denied in /home/karl-arne/public_html/privat/s.php on
line 42, referer: http://localhost/~karl-arne/privat/captcha_skjema.php
karl-arne@karlarne-Lenovo-Z50-70:~/public_html/privat$ sudo ps aux | grep
[sudo] password for karl-arne:
karl-ar+ 29230  0.0  0.0  17184  2212 pts/12   S+   16:30   0:00 grep
--color=auto httpd

I am using the example from