n =  10^60 + 7

  =  1000000000000000000000000000000000000000000000000000000000007

Trial division < 100000
 
n - 1  =  2 * 7 * 107 * composite

Pollard < 1000000 steps

n - 1  =  2 * 7 * 107 * 1679641 * 8255453 * c_44

unable to factor c_44:   primality proof FAILS.

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                            2ND  EXAMPLE
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n = 10^61 + 93

n - 1  =  10000000000000000000000000000000000000000000000000000000000093

trial divison < 100000

n - 1  =  2^2 * 3 * 2621 * 66037 * composite

Pollard < 1000000 steps

n - 1  = 2^2 * 3 * 2621 * 66037 * 3397567 * 1417086523103994077407747247360614547699381699


Miller-Rabin indicates that R = p_46 is prime and we do the Pocklington test:

y = 2^((n-1)/R) = 7177493323391307598818294406496164465181399154046999462613725 mod n

gcd(7177493323391307598818294406496164465181399154046999462613725 - 1, n) = 1

y^R = 1 mod n.

So now we know:  IF  R  IS PRIME, THEN  n  IS PRIME.

========================================
We now need to work with 

R = 1417086523103994077407747247360614547699381699

trial division < 100000

R - 1  = 2 * 59 * 919 * composite

Pollard < 1000000 steps

R - 1  = 2 * 59 * 919 * 385792526683 * 33872327408997649981423214143

Miller-Rabin indicates that R' = p_29 is prime and we do the Pocklington test:

y = 2^((R-1)/R') = 949429449846548911690145068225670508279718398 mod R

gcd(949429449846548911690145068225670508279718398 - 1, R) = 1

y^(R') = 1 mod R.

So now we know:  IF  R'  IS PRIME, THEN  R  IS PRIME.

=========================================================================
We now need to work with 

R' = 33872327408997649981423214143

trial division < 100000

R' - 1  = 2 * 3 * 137 * 3701 * 11134074833788477626361

Miller-Rabin indicates that R'' = p_23 is prime and we do the Pocklington test:

y = 2^((R'-1)/R'') = 1044414937576394936183462213 mod R'

gcd(1044414937576394936183462213 - 1, R') = 1

y^(R'') = 1 mod R'.

So now we know:  IF  R''  IS PRIME, THEN   R'   IS PRIME.

=============================================================

We now need to work with 

R'' = 11134074833788477626361

trial division < 100000

R'' - 1  = 2^3 * 3^2 * 5 * composite

Pollard < 1000000 steps

R'' - 1  = 2^3 * 3^2 * 5 * 201389 * 153573361253159

Miller-Rabin indicates that R''' = p_15 is prime and we do the Pocklington test:

y = 2^((R''-1)/R''') = 6823233954920652905481 mod R''

gcd(6823233954920652905481 - 1, R'') = 1

y^(R''') = 1 mod R''.

So now we know:  IF  R'''  IS PRIME, THEN  R''  IS PRIME.

==================================================================

We finally need to work with 

R''' = 153573361253159

trial division < 100000

R''' - 1  = 2 * 7 * 11 * 3547 * 281147341.

All factors are prime and the last one  q  exceeds sqrt(R''').
We do the Pocklington test:

y = 2^((R'''-1)/q) = 14965133413913 mod R'''

gcd(14965133413913 - 1, R''') = 1

y^q = 1 mod R'''.

So now we know  that  R'''  IS PRIME.

Backtracking shows that hence  R'',  hence  R',   hence R and hence  n  are primes.
And now we are done:

                      WE  HAVE  PROVED  THAT  n  IS  PRIME