// If x = n(3n−1)/2 => 2x = n(3n - 1) = 3n^2 - n => 3n^2 - n - 2x = 0
// Solving this quadratic equations yields:
// D = b^2 - 4ac = 1 - (4 * 3 * -2x) = 1 + 24x
// n = (1 - sqrt(1 + 24x))/6 or n = (1 + sqrt(1 + 24x))/6
// x and n are positive integers > 0, so n = (1 + sqrt(1 + 24x))/6
// So n is a pentagon number if 1 + 24x is a square number and sqrt(1 + 24x) == 5 mod 6.
public static bool IsPentagonalNumber(long integer)
{
var numberToCheck = 1 + 24 * integer;
return(NumberHelper.IsSquare(numberToCheck) && Math.Sqrt(numberToCheck) % 6 == 5);
}
public static long GetSolutionOfProblem92()
{
var limit = 10000000 - 1;
return((long)NumberHelper.NumberOfStartingNumbersForWhichSquareDigitChainEndsAt89(limit));
}
public static long GetSolutionOfProblem30()
{
var power = 5;
return(NumberHelper.GetIntegersWhichCanBeWrittenAsNthPowersOfTheirDigits(power).Sum());
}
public static long GetSolutionOfProblem62()
{
var desiredNumberOfCubicPermutations = 5;
return((long)NumberHelper.SmallestCubeWhichHasNCubicPermutations(desiredNumberOfCubicPermutations));
}
public static long GetSolutionOfProblem63()
{
return(NumberHelper.AllNDigitPositiveIntegersWhichAreNthPowers().Count);
}
public static long GetSolutionOfProblem58()
{
var desiredUpperBoundPercentagePrimes = 10;
return(NumberHelper.GetLengthOfNumberSpiralForWhichPercentageOfPrimesOnDiagonalsIsLowerThanLimit(desiredUpperBoundPercentagePrimes));
}
public static long GetSolutionOfProblem29()
{
var maxOfIntegersUsed = 100;
return(NumberHelper.GetListOfDistinctPowers(maxOfIntegersUsed).Count());
}
public static long GetSolutionOfProblem56()
{
var max = 99;
return(NumberHelper.MaximalDigitSum(max));
}
public static long GetSolutionOfProblem52()
{
return(NumberHelper.SmallestNumberSuchThat2345And6TimesTheNumberArePermutationsOfOriginalNumber());
}
public static long GetSolutionOfProblem28()
{
var sizeOfSpiralSides = 1001;
return(NumberHelper.GetSumOfDiagonalsNumberSpiral(sizeOfSpiralSides));
}
