A380031 Smallest integer of d digits, greater than 1 and not ending in 0, whose constant congruence speed is not yet constant at height d + 2.

5, 807, 81666295807, 81907922943, 161423787862411847003581666295807, 115161423787862411847003581666295807, 45115161423787862411847003581666295807, 44317662666830362972182803640476581907922943, 776138023544317662666830362972182803640476581907922943

Offset: 1

Author

Marco Ripà, Jan 10 2025

Keywords

• nonn
• base
• hard

Comments

The present sequence is a subsequence of A068407, but it is not a subsequence of A379906 (e.g., a(4) is not a term of A379906).

Although the congruence speed of any integer m > 1 not divisible by 10 is certainly stable at height m + 1 (for a tighter upper bound see "Number of stable digits of any integer tetration" in Links), this sequence contains infinitely many terms, implying the existence of infinitely many tetration bases of d digits whose congruence speed does not stabilize in less than d + 3 iterations (e.g., the congruence speed of 807, a 3-digit number, becomes constant only at height).

As a nontrivial example, the congruence speed of a(10) := 712222747129609220545115161423787862411847003581666295807 (a 57-digit number whose constant congruence speed is also 57) becomes stable at height 60, which exactly matches the mentioned tight bound, for the numbers ending in 2, 3, 7, or 8, of v_5(712222747129609220545115161423787862411847003581666295807^2 + 1) + 2, where v_5(...) indicates the 5-adic valuation of the argument.

The smallest integer of d digits whose constant congruence speed is not yet constant at height d + 3 is 435525708925199660525680385844696084258785712222747129609220545115161423787862411847003581666295807 (a 99-digit number whose congruence speed stabilizes at height 104 to its constant value of 101).

For any n >= 2, terms of this sequence derive from one digit 5 that appears in any of the two 10-adic solutions (- {5^2^k}_oo + {2^5^k}_oo) := ...2411847003581666295807 and (- {5^2^k}_oo - {2^5^k}_oo) := ...2803640476581907922943 of the fundamental 10-adic equation y^5 = y (see "The congruence speed formula" in Links). The only other candidate terms can arise from the remaining two symmetric 10-adic solutions ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 and ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193 of y^5 = y as particular patterns of 0s and 5 may occur in the corresponding (neverending) strings (e.g., '50...0').

Consequently, if n > 1 is given, a(n) is always congruent modulo 50 to 7 or 3.

Examples

			a(2) = 807 since the corresponding 10-adic solution of y^5 = y is ...61423787862411847003581666295807 where the rightmost digit 5 appears to the left side of a(2) itself, while no smaller numbers with the same feature are achievable by cutting the 10-adic integer ...30362972182803640476581907922943 (also one of the 15 solutions of the fundamental 10-adic equation y^5 = y) in correspondence of its rightmost digit 5.
		

References

• Marco Ripà, La strana coda della serie n^n^...^n, Trento, UNI Service, Nov 2011. ISBN 978-88-6178-789-6.

Links

• Marco Ripà, On the constant congruence speed of tetration, Notes on Number Theory and Discrete Mathematics, Volume 26, 2020, Number 3, Pages 245—260.
• Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
• Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024.
• Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
• Wikipedia, Tetration.

Crossrefs

Cf. A068407, A290372, A290373, A290374, A290375, A317905, A370211, A370775, A371129, A371671, A372490, A373387, A379906.

Formula

For any n > 1, a(n) corresponds to a cut on the right side of a digit 5 that appears inside one of the two strings (- {5^2^k}_oo - {2^5^k}_oo) := ...96579486665776138023544317662666830362972182803640476581907922943 and (- {5^2^k}_oo + {2^5^k}_oo) := ...84258785712222747129609220545115161423787862411847003581666295807, or even to a cut on the right side of a 5 belonging to rare digit-patterns consisting of juxtaposed 5 and trailing 0's occurring inside ({5^2^k}_oo + {2^5^k}_oo) := ...7196359523418092077057 or ({5^2^k}_oo - {2^5^k}_oo) := ...7588152996418333704193.