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#define speakerPin 8

unsigned long lastClick;

void setup() {
  // put your setup code here, to run once:
   pinMode(speakerPin,OUTPUT);
   lastClick = micros();   
}


/* initialize with any 32 bit non-zero  unsigned long value. */
#define LFSR_INIT  0xfeedfaceUL
/* Choose bits 32, 30, 26, 24 from  http://arduino.stackexchange.com/a/6725/6628
 *  or 32, 22, 2, 1 from 
 *  http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf
 *  or bits 32, 16, 3,2  or 0x80010006UL per http://users.ece.cmu.edu/~koopman/lfsr/index.html 
 *  and http://users.ece.cmu.edu/~koopman/lfsr/32.dat.gz
 */  
#define LFSR_MASK  ((unsigned long)( 1UL<<31 | 1UL <<15 | 1UL <<2 | 1UL <<1  ))

unsigned int generateNoise(){ 
  // See https://en.wikipedia.org/wiki/Linear_feedback_shift_register#Galois_LFSRs
   static unsigned long int lfsr = LFSR_INIT;  /* 32 bit init, nonzero */
   /* If the output bit is 1, apply toggle mask.
                                    * The value has 1 at bits corresponding
                                    * to taps, 0 elsewhere. */
                                
   if(lfsr & 1) { lfsr =  (lfsr >>1) ^ LFSR_MASK ; return(1);}
   else         { lfsr >>= 1;                      return(0);}
}


void loop() {
      /* ... */
      if ((micros() - lastClick) > 50 ) { // Changing this value changes the frequency.
        lastClick = micros();
        digitalWrite (speakerPin, generateNoise());
      }
     
}

Where the mask constants are from the 8.txt and 16.txt files at http://users.ece.cmu.edu/~koopman/lfsr/index.html

#define speakerPin 8

unsigned long lastClick;

void setup() {
  // put your setup code here, to run once:
   pinMode(speakerPin,OUTPUT);
   lastClick = micros();   
}


/* initialize with any 32 bit non-zero  unsigned long value. */
#define LFSR_INIT  0xfeedfaceUL
/* Choose bits 32, 30, 26, 24 from  http://arduino.stackexchange.com/a/6725/6628
 *  or 32, 22, 2, 1 from 
 *  http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf
 *  or bits 32, 16, 3,2  or 0x80010006UL per http://users.ece.cmu.edu/~koopman/lfsr/index.html 
 *  and http://users.ece.cmu.edu/~koopman/lfsr/32.dat.gz
 *  or generate a one with https://github.com/hayguen/mlpolygen
 */  
#define LFSR_MASK  ((unsigned long)( 1UL<<31 | 1UL <<15 | 1UL <<2 | 1UL <<1  ))

unsigned int generateNoise(){ 
  // See https://en.wikipedia.org/wiki/Linear_feedback_shift_register#Galois_LFSRs
   static unsigned long int lfsr = LFSR_INIT;  /* 32 bit init, nonzero */
   /* If the output bit is 1, apply toggle mask.
                                    * The value has 1 at bits corresponding
                                    * to taps, 0 elsewhere. */
                                
   if(lfsr & 1) { lfsr =  (lfsr >>1) ^ LFSR_MASK ; return(1);}
   else         { lfsr >>= 1;                      return(0);}
}


void loop() {
      /* ... */
      if ((micros() - lastClick) > 50 ) { // Changing this value changes the frequency.
        lastClick = micros();
        digitalWrite (speakerPin, generateNoise());
      }
     
}

Where the mask constants are from the 8.txt and 16.txt files at http://users.ece.cmu.edu/~koopman/lfsr/index.html

The https://github.com/hayguen/mlpolygen program can search for maximum length feedback polynomials of various bit sizes.

For an 8 bit implementation (faster than 32 bit, slower than 16, but only a cycle of 2^8) try:

#define LFSR_MASK 0x8e 
static byte lfsr = LFSR_INIT;  /* 16 bit init, nonzero */

Written in a more compact form:

#define speakerPin 8

#define LFSR_INIT 0xfeed /* non-zero seed value for generateNoise() */
#define LFSR_MASK 0x8016 /* magic number from http://users.ece.cmu.edu/~koopman/lfsr/ */

unsigned long lastClick;

void setup() {
  // put your setup code here, to run once:
   pinMode(speakerPin,OUTPUT);
   lastClick = micros();   
}


unsigned int generateNoise(){ 
   // Return 1 bit of noise using a Galois Linear Feedback Shift Register
   // See https://en.wikipedia.org/wiki/Linear_feedback_shift_register#Galois_LFSRs

   static uint16_t lfsr = LFSR_INIT; 
                                
   if(lfsr & 1) { lfsr =  (lfsr >>1) ^ LFSR_MASK ; return(1);}
   else         { lfsr >>= 1;                      return(0);}
}


void loop() {
      /* ... */
      if ((micros() - lastClick) > 50 ) { // Changing this value changes the frequency.
        lastClick = micros();
        digitalWrite (speakerPin, generateNoise());
      }
     
}

For an 8 bit implementation (faster than 32 bit, slightly slower than 16, but only a cycle of 2^8) try:

#define LFSR_MASK 0x8e  

static byte lfsr = LFSR_INIT;  /* 8 bit init, nonzero */

For a 16 bit implementation (faster, but only a cycle of 2^16) try:

#define LFSR_MASK 0x8016 
static uint16_t lfsr = LFSR_INIT;  /* 16 bit init, nonzero */

For an 8 bit implementation (faster than 32 bit, slower than 16, but only a cycle of 2^8) try:

#define LFSR_MASK 0x8e 
static byte lfsr = LFSR_INIT;  /* 16 bit init, nonzero */

Where the mask constants are from the 8.txt and 16.txt files at http://users.ece.cmu.edu/~koopman/lfsr/index.html