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A256580
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Number of quadruples (x, x+1, x+2, x+3) with 1 < x < p-3 of consecutive integers whose product is 1 mod p.
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3
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0, 0, 0, 1, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 4, 0, 4, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 4, 2, 0, 2, 0, 0, 2, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 4, 2, 2, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 4, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 4, 0, 2, 2, 0, 0, 4, 4, 0, 4, 2, 0, 0
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OFFSET
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1,7
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COMMENTS
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If "quadruples" is changed to "pairs" we get A086937 (for the counts) and A038872 (for the primes for which the count is nonzero).
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LINKS
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FORMULA
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|T| where T = {x|x*(x+1)*(x+2)*(x+3) == 1 mod p, p is prime, 1 < x < p-3}.
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EXAMPLE
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p=7, x_1=2, 2*3*4*5 == 1 (mod 7), T={2}, |T|=1;
p=17, x_1=2, 2*3*4*5 == 1 (mod 17), x_2=12, 12*13*14*15 == 1 (mod 17), T={2,12}, |T|=2;
p=23, x_1=5, 5*6*7*8 == 1 (mod 23), x_2=15, 15*16*17*18 == 1 (mod 23), x_3=19, 19*20*21*22 == 1 (mod 23), T={5,15,19}, |T|=3.
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PROG
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(R)
library(numbers); IP <- vector(); t <- vector(); S <- vector(); IP <- c(Primes(1000)); for (j in 1:(length(IP))){for (i in 2:(IP[j]-4)){t[i-1] <-as.vector(mod((i*(i+1)*(i+2)*(i+3)), IP[j])); Z[j] <- sum(which(t==1)); S[j] <- length(which(t==1))}}; S
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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