__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, 2

^{r}, 2

^{r-1}, 2

^{r-2}, ... , 1] then a Collatz trap n is:

n = (2

^{r}+ 1) + ((2^{r}- 1) - m)**------------------------------------------------------------------------------------------------------------------------**

__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 m

^{2}= 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 2s

^{2}= ((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 k**

^{r}= 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 k

^{s}= 1 mod n so k^{s+1}= k*(k^{s}) = k*1= k mod n
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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 k

^{k}= 1 mod n then k is a lonely power mod n

**Claim:**If r is the smallest integer such that k

^{r}= 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 k

^{2}= 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 k

_{1}is at a distance d from the (((n-1)/2)+1)th element then there exists another selfie square k

_{2}= (((n-1)/2)+1) + d

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__Lonely Squares__

**Definition:**Given two distinct odd primes p and q and an integer

**< p*q such that GCD(**

*a***, p*q) = 1 and**

*a*

*a*^{2}= 1 mod p*q then

**a**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 = 2k mod 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 [2

^{0},2

^{1}, 2

^{2}, 2

^{3},..., 2

^{k}] but b

^{r/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 d

^{s}mod p*q is a lonely square then there is another lonely square c where:

c = p*q - d

^{s}