Solves the max decimal expansion length from 1 to N in a general way using the discrete logarithm. O(n^4)

max_decimal_expansion length.cs

using System;

namespace Test
{
	class RepeatingDecimalLength
	{
		// Solves the discrete logarithm problem in O(n^3). See https://arxiv.org/ftp/arxiv/papers/0912/0912.2269.pdf
		// Returns k that solves for x^k = y (mod p)
		public static UInt64 DiscreteLogarithm(UInt64 x, UInt64 y, UInt64 p)
		{
			UInt64 lv_lng_x = x;
			UInt64 lv_lng_y = y;
			UInt64 lv_lng_p = p;
			UInt64 lv_lng_vx1 = lv_lng_x;
			UInt64 lv_lng_vx2 = 0;
			UInt64 lv_lng_vy1 = lv_lng_y;
			UInt64 lv_lng_vy2 = lv_lng_vy1;
			UInt64 lv_lng_v1 = 1;
			UInt64 k = 0;

			for (System.UInt64 i = 0; i < lv_lng_p; i++)
			{
				lv_lng_vx2 = 0;

				for (System.UInt64 j = 0; j < lv_lng_x; j++)
				{
					lv_lng_vx2 += lv_lng_vx1;
				}

				lv_lng_vx1 = lv_lng_vx2;
				lv_lng_v1 += 1;

				while (lv_lng_vx1 > lv_lng_p)
				{
					lv_lng_vx1 -= lv_lng_p;
				}

				if (lv_lng_vy2 == lv_lng_vx1)
				{
					k = lv_lng_v1;
					break;
				}
			}

			return k;
		}

		public static void Main(string[] args)
		{
			// Multiplicative Order (Wolfram Alpha)
			// The multiplicative order of 10 mod an integer n relatively prime
			// to 10 gives the period of the decimal expansion of the reciprocal
			// of n (Glaisher 1878, Lehmer 1941).
			UInt64 k = 0;
			int num = 0;

			for (int i = 0; i < 1000; i++)
			{
				var result = DiscreteLogarithm(10, 1, (UInt64)i);
				if (result > k)
				{
					k = result;
					num = i;
					Console.WriteLine("{0} has {1} decimal expansion length", num, k);
				}
			}

			Console.ReadLine();
		}
	}
}