A Simple Password Puzzle. Or Maybe Not

Setting the scene

We have five numbers $alpha,beta,gamma,delta,epsilon$ satisfying seven conditions. These numbers must all be non-negative integers (we assume). I tried assuming they are all single digits (at most $9$), but got a contradiction as shown in the first revision of this answer, so we know $gammageq10$.

Step-by-step deduction

• Firstly, can any of them be zero?

• The password is prime, so

• Substituting condition 3 into condition 1, we find

• Going back to condition 6, we notice

• Now conditions 4 and 6 are used up, and condition 5 gives

Note that after fixing $beta$ and $epsilon$, the others are completely determined:

Let's now just try possibilities starting from the smallest:

• If $beta=3$, then we must have

• If $beta=4$, then we must have

• If $beta=5$, then we must have

Final solution for the password

Answer:

Checking

Partial explanation (to be continued):

OP's Answer

The answer is, as shown by @MacGvyer88,

but they did not provide logical deduction, so here it is.

@RandAlThor has proven that a five-digit number is impossible as the answer. Yet it is mentioned in the clarifications that the length is 5. Therefore it may be a

Well, intrinsically, you can say that

but I will try to deduct this on the approach that we only know that it is a five-letter word and that A1Z26 is used.

From (1) the max of LHS is 26+25+25 = 76 since gamma is unique largest. In other words beta <= floor(76/11) = 6.

Therefore beta is

And in second position it is very likely a

Notice condition 6 regarding duplicates.

From (2) gamma isn't a duplicate.

From (3) we know that alpha ≠ epsilon.

Based on all this we know that either beta or delta is one of the duplicates.

(5) shows that beta ≠ epsilon and delta ≠ epsilon, so epsilon is out.

We now have alpha, beta and delta being possible duplicates.

(3) shows that alpha ≠ beta

Therefore either alpha = delta or beta = delta.

Given beta = A or E most likely, we now have

And this should be enough, given (7) for you to make out

---

I am challenged to prove why beta is A or E. First of all I said "most likely". Secondly, here it is if you need proof.