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한국고등과학원 세미나 



[GSNT] Universal sums of generalized octagonal numbers 

An integer of the form P8(x) = 3x2 ? 2x for some integer x is called a generalized octagonal number. A quaternary sum Φa,b,c,d(x, y, z, t) = aP8(x) + bP8(y) + cP8(z) + dP8(t) of generalized octagonal numbers is called universal if Φa,b,c,d(x, y, z, t) = n has an integer solution x, y, z, t for any pos itive integer n. In this talk, we show that if a = 1 and (b,c,d) = (1,3,3), (1, 3, 6), (2, 3, 6), (2, 3, 7) or (2, 3, 9), then Φa,b,c,d(x, y, z, t) is universal. These were conjectured by Sun. We also give an effective criterion on the universal ity of an arbitrary sum a1P8(x1) + a2P8(x2) + · · · + akP8(xk) of generalized octagonal numbers, which is a generalization of “15theorem” of Conway and Schneeberger. This is a joint work with ByeongKweon Oh. 






