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The paper presents a practical method for factoring an arbitrary N by representing N or \N by one of at most three quadratic forms: XAf = s2 -Dy2, X = 1, -1, 2, D = -1, ±2, ±3, ±6. These three forms appropriate to N, together with inequalities for y, are given for all N prime to 6. Presently available sieving facilities make the method quite effective and economical for numbers N having 20 to 25 digits. Four examples arising from aliquot series are discussed in detail.doi:10.1090/s0025-5718-1974-0342458-5 fatcat:oqzo2atrejh63ijhbu6mu6uxjq