This post is about the Wieferich primes, 1093 and

3511.

(1) Concerning the two in concert producing a prime P

(2) Concerning what I have found to be an interesting

composite for P+1 and possibilities for primes close

to it

(3) Concerning 3511 itself, a Wieferich prime.

(1) Let f(x,y) = x^2 - y^2 -2xy.

f(3511,1093) = 7294949 and is prime P.

(2) 7294950 = 2*3*3*5*5*13*29*43 which made it easy to

factorise into its 8 factors, of which 5 are less than

(P+1)^(1/8) and all less than (P+1)^(1/4). This fact

seem to me to make 7294950 fairly interesting as only

6435 previous composites have the property of having 8

prime factors, one in more than a thousand.

(3) f(79,26) = 3511 itself remarkable as 79 = 3*26+1

which makes it a "near" Lucas series with its

characteristic second term 79 = 1mod3 and 1mod(the

first term viz. 26 in this case).

(I have postulated that any prime 1mod10 or -1mod10 is

always expressible with x > 3y in f(x,y), and that x

is the second term of a Fibonacci series T(1) = y,

T(2) = x.)

I have more interesting material but unless this

creates interest, I will spare you it.

John McNamara

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