how to determine the max user process value?

Question

Which value is correct?(or they are all correct, but which one will take effect?)

$ cat /proc/sys/kernel/pid_max 
32768
$ ulimit -a |grep processes
max user processes              (-u) 77301
$ cat /proc/1/limits |grep processes
Max processes             77301                77301                p

Answer 1

All values is correct and have different meanings./proc/sys/kernel/pid_max is maximum value for PID, ulimit -u is maximum value for number of processes.

From man 5 proc:

/proc/sys/kernel/pid_max (since Linux 2.5.34)
              This  file  specifies the value at which PIDs wrap around (i.e.,
              the value in this file is one greater  than  the  maximum  PID).
              The  default  value  for  this  file, 32768, results in the same
              range of PIDs as on earlier kernels.  On 32-bit platforms, 32768
              is  the  maximum  value for pid_max.  On 64-bit systems, pid_max
              can be set to any value up to 2^22 (PID_MAX_LIMIT, approximately
              4 million).

From man bash:

ulimit [-HSTabcdefilmnpqrstuvx [limit]]
              .....
              -u     The maximum number of processes available to a single user
              .....

Note

When a new process is created, it is assigned next number available of kernel processes counter. When it reached pid_max, the kernel restart the processes counter to 300. From linux source code, pid.c file:

....
#define RESERVED_PIDS       300
....
static int alloc_pidmap(struct pid_namespace *pid_ns)                           
{                                                                               
    int i, offset, max_scan, pid, last = pid_ns->last_pid;                      
    struct pidmap *map;                                                         

    pid = last + 1;                                                             
    if (pid >= pid_max)                                                         
        pid = RESERVED_PIDS;