Puzzle: The powers 2^n and 5^n have the same first digit. Which is it? Finding examples, where the first digits of 2^n and 5^n are the same. 2^n * 5^n = 10^n Case 1: n = 0 the first digit is 1. Case 2: n > 0 We are looking for approximations of 2^n = 10^(m + 1/2) 2m + 1 p ------ = - = log10(2) = 0.3010299956639811952137388947244930267681898814621 2n q continued fraction of log10(2) is [0, 3, 3, 9, 2, 2, 4, 6, 2, 1, 1, 3, 1, 18, 1, 6, 1, 2, 1, 1, 4, 1, 42, 6, 1,] p q q/2 log10(2) -------------------------------------------------------------------- 3 1 3 3 3 10 1.505149978 9 28 93 2 59 196 29.500939575 2 146 485 4 643 2136 321.500035369 6 4004 13301 2 8651 28738 4325.500007695 1 12655 42039 1 21306 70777 3 76573 254370 38286.499998523 1 97879 325147 18 1838395 6107016 919197.499999931 1 1936274 6432163 6 13456039 44699994 6728019.499999992 1 15392313 51132157 2 44240665 146964308 1 59632978 198096465 1 103873643 345060773 4 475127550 1578339557 1 579001193 1923400330 42 24793177656 82361153417 6 149338067129 496090320832 1 174131244785 578451474249 4 845863046269 2809896217828 2 1865857337323 6198243909905 3 6443435058238 21404627947543 1 8309292395561 27602871857448 q=10 2^5 = 32 5^5 = 3125 q=196 2^98 = 316912650057057350374175801344 5^98 = 315544362088404722164691426113114491869282574043609201908111572265625 q=2136 2^1068 = 3.16253520792 * 10^321 5^1068 = 3.16202013338 * 10^746 q=28738 2^14369 = 3.16233369657 * 10^4325 5^14369 = 3.16222162475 * 10^10043