User blog comment:Allam948736/Last digits of tetrations - proof/@comment-35470197-20200212031655

On the first issue, your statement is a proposition on x, y, b, and d without quantifiers. Then you need to appropriately quantify it. Otherwise, you can shift the goalpost by saying something like "no, I obviously assumed that x is a prime number comprime with b, y is greater than 10^100, and d satisfies a secret property" after you are given a counterexample.
 * Write down a precise statement with full quantifications.
 * If you want to prove a statement X, then deduce X from valid arguments.

On the second issue, you wrote an argument on a, b, c, and d. How you deduced a statement on x, y, b, and d? Specify the way to eliminate the qunatifier to deduce your statement on x, y, b, and d from your arguments on a, b, c, and d.

On the arguments themselves, there are several errors.
 * 1) You have not clarified your statement. You just defined s_y. Is your statement the quantification of "s_y coincides with the last d digits of x↑↑y in base b"? Clearly fix your statement with full qunatification.
 * 2) The first occurrence of s_n + 1 is perhaps a typo of s_{n+1}. This is just a minor mistake.
 * 3) The condition "c cannot be expressed as m^n for unique positive integers m and n" is a little abiguous. You meant the negation of "there uniquely exists a pair (m,n) of positive integers such that c = m^n", right?
 * 4) The reasoning "ruling out any possibilities of a^b mod c" is quite ambiguous. Although it is quite easy for readers to justify the argument, it does not mean the original argument is a valid proof. Write down the argument precisely.
 * 5) The reasoning "the number of possibilities remaining for the last base-c digit of a^b is φ(c) by definition of Euler's totient function" is not correct, because you need to check a^b mod c actually runs through all values in {0,…,c-1} coprime to c. Although it is quite easy for readers to justify the argument, it does not mean the original argument is a valid proof. Please write down the argument precisely.
 * 6) The reasoning "Since q divides a^b for any positive integer value b" is invalid, because you have fixed a positive integer b. Do not redefine or requantify b.
 * 7) The statement "the period of the last base-c digit can be no more than ～" does not make sense. What is "the period of the last base-c digit"? What is the last base-c digit? Is it the last base-c digit for any positive integer?
 * 8) Clarify the definition of the period in this context, because it is quite ambiguous here.
 * 9) Write down the proof of the estimation by φ(c/q).
 * 10) The sentence "I will now prove the statement for more base-c digits." does not make sense. Have you finished to prove your statement for the case d = 1? By the argument without the occurrence of x and y? No. You have not verified your statement.
 * 11) The estimation "leaving φ(c^2)=φ(c)*c possibilities for the last 2 base-c digits of a^b." is not proved by the same reason for a^n mod c above.
 * 12) Write down the proof of the estimation by φ((c/q)^2).
 * 13) The sentence "We can continue proving this for more digits, but now for the good part" does not make sense. What does "this" mean? Are you talking about your statement on x, y, b, and d? Have you finished to prove your statement for small digits? Without the occurrence of x and y? No way.
 * 14) Write down the proof of the estimation by φ(c^d).
 * 15) The reasoning "since the period ～ can be no more than φ(c^d)" is wrong, because the equality (a^{γ-δ} - a^γ) mod c^d = 0 for any positive integers γ and δ satisfying δ≦γ and δ ≦ φ(c^d) does not imply (a^{γ-φ(c^d)}-a^γ) mod c^d = 0 for any positive integer γ greater than or equal to φ(c^d).
 * 16) The sentence "meaning the last digits of a↑↑b in base c coincide with those of a^{a↑↑(b-1) mod lcm(c^d,φ(c^d))}" is wrong, because it does not mean the result.
 * 17) The reasoning "Therefore, the last digits of tetrations can be computed using the method I propose." is wrong. You have never deduced your statement on x, y, b, and d. Moreover, what you wrote is the computation of a^{a↑↑(b-1) mod lcm(c^d,φ(c^d))} mod c^d, which is not directly related to x^{x↑↑(y-1) mod b^d} mod lcm(b^d,φ(b^d)). Simple elimination of universal quantifiers by setting x=a, y=b(in the first equality), b(in the second equality)=c, and d(in the second equality) = d(in the first equality) does not given an obvious implication from the first equality to the second equality. If you are stating x^{x↑↑(y-1) mod b^d} mod lcm(b^d,φ(b^d)) is equivalent to x^{x↑↑(y-1) mod lcm(b^d,φ(b^d))} mod b^d under an appropriate condition, then you need to write down a proof.

Also, what you want to show looks just a very weak portion of the statement in the article. Why are you trying to verify it as if it were your own new investigation by writing " I proposed another method for computing last digits"? Are you really understanding the article? Or do you have a hidden statement, which is not written in your blog post?