However, the two underlined digits should be 13. Bell) computed the first 31 digits and the last 12 digits of the smallest total number of cattle and found them to beħ7602714064868182695302328332 09. The club's three members (Edmund Fish, Geo. “S ince it has been calculated that it would take the work of a thousand men for a thousand years to determine the complete number, it is obvious that the world will never have a complete solution.”Īmthor's calculations were continued by an ad hoc group called the Hillsboro Mathematical Club (Hillsboro, Illinois, USA) in the years 1889 to 1893. ()Īmthor found that the smallest integers r and m that satisfy this equation result in a total number of cattle expressed by an integer with 206,545 digits that begins with the digits 776. A partial solution to this problem was given in The problem now is to find positive integers r and m that satisfy this last equation. Using our previous condition for the allowable values of k, this becomes By using the formula for the sum of the first m integers, we can also characterize triangular numbers as those numbers of the form Where triangular numbers are numbers of the form when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure. The second additional condition, which further restricts the allowable value of k, states For the left-hand side of this equation to be a square number, it follows that k must be of the form Where, in the last equation, the number 17,826,996 has been expressed as a product of prime numbers. The most direct interpretation of this condition is thatġ0,366,482 k + 7,460,514 k = a square number When the while bulls mingled their number with the black, they stood firm, equal The second part of the cattle problem imposes two additional conditions that restrict the possible values of k beyond k = 1, 2, 3. The first part of the cattle problem has infinitely many solutions given by
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