Sylver Coinage: The Progression Hypothesis

The Hypothesis

Following Roberts, let a[0], a[1], ..., a[s] be an increasing arithmetic progression with successive difference d. Assume that a[0] and d are coprime and greater than 1. Considering S = {a[0],...,a[s]} as a position in Sylver Coinage, we can also assume that s<a[0], since for any k, a[k+a[0]] = a[k] + da[0], so that a[0] and a[k] together eliminate a[k+a[0]].

Let t = t(S). The value of t is given by Roberts's Formula. This is the Progression Hypothesis:


Let S={2,9}. Since a[0]=2, S is a quiet ender.

Let S={3,7,11}. Since a[0]=3 and s=2, S is a non-quiet ender.

Let S={5,6,7,8,9}. Since a[0]>3 and s=a[0]-1, S is not an ender. Here t=4, s=4, and d=1; and 4 fails to be eliminated by the one multiple of 1 that does not divide s: 3.

Let S={7,11,15,19,23,27,31}. Again a[0]>3 and s=a[0]-1, so S is not an ender. Here t=24, s=6, and d=4; since neither 4 nor 5 divides 6, 24 fails to be eliminated by 16 and 20.

Let S={6,13,20,27}. In this case a[0]>3 and s<a[0]-1. We find that t=41, s=3, and d=7. Since s leaves a remainder of 1 when divided into a[0]-2, 41 fails to be eliminated by (41-1*7)=34.

Let S={8,11,14,17,20,23,26}. Again a[0]>3 and s<a[0]-1. Here t=29, s=6, and d=3. Since s | a[0]-2, S is a quiet ender.

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Col. G. L. Sicherman [ HOME | MAIL ]