Glossary

Terms related to The Collatz Conjecture

Definition: An exit point in relation to the Collatz conjecture is the last odd integer reached before reaching a power of 2.

Example: Let s = 10 then the Collatz exit point is 5 since 10/2 = 5 and 3*5 + 1 = 16, which is a power of 2.

The first few Collatz exit points are 5,21,85,341,1365,2731,5461,21845

Definition: An exit path in relation to the Collatz conjecture is the last few integers reached in a Collatz trajectory starting from the Collatz exit point.

Example: Let s  = 10 then its Collatz exit path is 5,16,8,4,2,1

Note: In my previous blog entry I used the term "Collatz path" but I decided that "Collatz exit path" is a more descriptive term.

Definition: A Collatz trap is an odd integer n such that the last few powers of (n+1)/2 correspond to a Collatz exit path.

Example: Let n = 27, the last few powers of 14 mod 27 are 5, 16, 8, 4, 2, 1, which correspond to the Collatz exit path of 10.

The first few Collatz traps are: 27, 107, 427, 1707, 6807


Claim: Given a Collatz exit path [m, 2r, 2r-1, 2r-2, ... , 1] then a Collatz trap n is:

n = (2r + 1) + ((2r - 1) - m)

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Skewy

Definition: The skewy is the smallest square root of (((n-1)/2)+1)2  modulo n when n is the product of distinct prime numbers.

Claim: If n is the product of two distinct primes p < q then (((n-1)/2)+1)2 has exactly 4 square roots.

Claim: If n is the product of two distinct primes p and q, s is the skewy mod n, and

m = ((((n-1)/2)+1) +  s) mod n

then m is either equal to p or q or m is a small multiple of p or q such that m2 = m mod n

Claim: If s is the skewy mod n then 2*s is a lonely square.
In other words, if s is the smallest square root of (((n-1)/2)+1)2 then (2s)2 = 1 mod n

Claim: If s is the skewy mod n then 2s2 = ((n-1)/2) + 1

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Rhythm of an element

Definition: The rhythm of k mod n is the smallest integer r > 1 such that kr = k mod n

Claim: The rhythm of k mod n is the number of unique integers that are multiples of k mod n.

Proof. Given an integer k, then the next integer is k*k, and each subsequent integer is k*(k*k... mod n) until (k*k... mod n) = 1, therefore the rhythm of k mod n counts the number of unique integers that are multiples of k mod n.

Claim: If GCD(k,n) = 1 and s is the order of k mod n then the rhythm r of k mod n is r = s+1

Proof: If s is the order of k mod n then ks = 1 mod n so ks+1 = k*(ks) = k*1= k mod n

Claim: If r is the rhythm of k mod n and k divides n then kr-1 is a selfie square.

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Tough Primes

Definition: Tough primes are primes q of the form 2p + 1 where p is a Sophie Germain prime such that q cannot also be represented as 8n + 7 for some n in Z.

Claim The order of 2 mod n when n is a tough prime, or the product of tough primes is precisely phi(p*q)/2

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Lonely Powers

Definition: Given n, k such that n < k, then k is a lonely power mod n if k to the power of k is congruent to 1 mod n. In other words, if kk = 1 mod n then k is a lonely power mod n

Claim: If r is the smallest integer such that kr = 1 mod n and r divides k then k is a lonely power.

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Selfie Squares

Definition: Given k < n, then k is a selfie square mod n if k2 = k mod n

Claim: If n is the product of distinct odd primes then there are (at least) 2 non-trivial selfie squares mod n such that if the first non-trivial selfie square k1 is at a distance d from the (((n-1)/2)+1)th element then there exists another selfie square k2 = (((n-1)/2)+1) + d

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Lonely Squares

Definition: Given two distinct odd primes p and q and an integer a < p*q such that GCD(a, p*q) = 1 and a2 = 1 mod p*q then is a lonely square mod p*q.

Claim: If 2 has order k mod p*q where p and q are distinct odd primes and k is an even integer then (2k/2) is a lonely square.

Proof:  Since k is the order of 2 then 2k= 1 mod p*q
Since k is an even integer then k/2 is also an integer and so:
(2k/2)2 mod p*q = (2(k/2)*2) mod p*q = 2mod p*q = 1 mod p*q

Claim: If 2k/2 =  (p*q - 1) mod p*q where k is the order of 2 mod p*q and it is even and p, q > 2 then there is at least one other small prime b of order r where r is even such that b does not belong to the set [20,21, 22, 23,..., 2k] but  br/2  mod p*q is a lonely square.


Claim: If p, q > 2 are two distinct primes then there are at least 4 lonely squares.


Claim: If ds mod p*q is a lonely square then there is another lonely square c where:

c = p*q - ds
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