for integers m and n, i've been factoring (over the integers) the polynomials: P = (x^n-1) / (x-1) - m obviously when m = (k^n-1)/(k-1) for some integer k, there is a linear factor. i'm interested in the cases when P factors without a linear factor. i've only found 6 cases: (x^8-1)/(x-1) - 3 = (x^3 + x – 1)(x^4 + x^3 + x + 2) (x^14-1)/(x-1) - 4 = (x^3 + x^2 – 1)(x^10 + x^8 + x^7 + 2x^5 + x3 + 2x^2 – x + 3) (x^5-1)/(x-1) + 11 = (x^2 + 3x + 4)(x^2 – 2x + 3) (x^9-1)/(x-1) + 19 = (x^4 – x^3 + x^2 – 3x + 4)(x^4 + 2x^3 + 2x^2 + 4x + 5) (x^11-1)/(x-1) - 23 = (x^2 + x + 2)(x^8 – x^6 + 2x^5 + x^4 – 4x^3 + 3x^2 + 6x – 11) (x^6-1)/(x-1) - 56 = (x^2 – x + 5)(x^3 + 2x^3 – 2x – 11) is there some rhyme or reason for these, or are they just random? erich