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The algorithm Given a natural number $n$ define a procedure as follows:

• Generate a list of primes upto and possibly including, $n$
• Assign $Z = n$
• If $Z > 0$, subtract the largest prime from list which we haven't considered yet. Otherwise, add it to $Z$. If $n$ is prime, it is assumed accounted for by the first step.
• Repeat until all primes have been considered.

For example, take $25$. The list of primes would be $2, 3, 5, 7, 11, 13, 17, 19, 23$. Subtracting $23$ from $Z=25$, we get $Z=2$. Next, we get $Z=2-19= -17$. And so on. Consequently, $Z$ assumes the values $25, 2, -17, 0, 13, 2, -5, 0, 3, 1$.

_Note: Only an example. The conjecture as stated deals with applying the algorithms on primes. However, other numbers also seem to exhibit interesting patterns._

The Pattern

• Beginning at $3$ and every other prime thereafter, following the algorithm always (seems to) land us at $1$.
• For the rest of the primes, $Z$ has a final value of either $0$ or $2$.

The problem

Please read @alex.jordan's answer as he cleverly limits the range of values $Z$ can reach (say, terminal Z, or $Z_t$) to $\{-1,0,1,2\}$. As a result, the problem is now reduced to:

• For any prime number, prove that $-1$ cannot be a terminal.

Also being discussed: here

The algorithm Given a natural number $n$ define a procedure as follows:

• Generate a list of primes upto and possibly including, $n$
• Assign $Z = n$
• If $Z > 0$, subtract the largest prime from list which we haven't considered yet. Otherwise, add it to $Z$. If $n$ is prime, it is assumed accounted for by the first step.
• Repeat until all primes have been considered.

For example, take $25$. The list of primes would be $2, 3, 5, 7, 11, 13, 17, 19, 23$. Subtracting $23$ from $Z=25$, we get $Z=2$. Next, we get $Z=2-19= -17$. And so on. Consequently, $Z$ assumes the values $25, 2, -17, 0, 13, 2, -5, 0, 3, 1$.

_Note: Only an example. The conjecture as stated deals with applying the algorithms on primes. However, other numbers also seem to exhibit interesting patterns._

The Pattern

• Beginning at $3$ and every other prime thereafter, following the algorithm always (seems to) land us at $1$.
• For the rest of the primes, $Z$ has a final value of either $0$ or $2$.

The problem

Please read @alex.jordan's answer as he cleverly limits the range of values $Z$ can reach (say, terminal Z, or $Z_t$) to $\{-1,0,1,2\}$. As a result, the problem is now reduced to:

• For any prime number, prove that $-1$ cannot be a terminal.

Also being discussed: here

Which has been puzzling me over the last year.

The algorithm Given a natural number $n$ define a procedure as follows:

• Generate a list of primes upto and possibly including, $n$
• Assign $Z = n$
• If $Z > 0$, subtract the largest prime from list which we haven't considered yet. Otherwise, add it to $Z$. If $n$ is prime, it is assumed accounted for by the first step.
• Repeat until all primes have been considered.

For example, take $25$. The list of primes would be $2, 3, 5, 7, 11, 13, 17, 19, 23$. Subtracting, $23$ from $Z=25$, we get $Z=2$. Next, we get $Z=2-19= -17$. And so on.

  Consequently, $Z$ assumes the values $25, 2, -17, 0, 13, 2, -5, 0, 3, 1$.

The Pattern

• Beginning at $3$ and every other prime thereafter, following the algorithm always (seems to) land us at $1$.
• For the rest of the primes, $Z$ has a final value of either $0$ or $2$.

The problem

I would like to know, for instance, what mathematical tools would be best suited to attempting a proof? Perhaps all this is simply a consequence of something more powerful, like the Goldbach conjecture?

Maybe what I'm really expecting is some ingenious proof I can't understand, or a contradiction. I feel I've beaten this idea just about dead and can't go further with my understanding of Mathematics.

Updates

Please read @alex.jordan's answer as he cleverly limits the range of values $Z$ can reach (say, terminal Z, or $Z_t$) to $\{-2,-1,0,1,2\}$. The missing piece to the puzzle seems to be a proof that no prime number can have $Z_t=-1$. This appears to be extendable to $Z_t(n)=-1$ iff n is even.

The algorithm Given a natural number $n$ define a procedure as follows:

• Generate a list of primes upto and possibly including, $n$
• Assign $Z = n$
• If $Z > 0$, subtract the largest prime from list which we haven't considered yet. Otherwise, add it to $Z$. If $n$ is prime, it is assumed accounted for by the first step.
• Repeat until all primes have been considered.

For example, take $25$. The list of primes would be $2, 3, 5, 7, 11, 13, 17, 19, 23$. Subtracting $23$ from $Z=25$, we get $Z=2$. Next, we get $Z=2-19= -17$. And so on. Consequently, $Z$ assumes the values $25, 2, -17, 0, 13, 2, -5, 0, 3, 1$.

_Note: Only an example. The conjecture as stated deals with applying the algorithms on primes. However, other numbers also seem to exhibit interesting patterns._

The Pattern

• Beginning at $3$ and every other prime thereafter, following the algorithm always (seems to) land us at $1$.
• For the rest of the primes, $Z$ has a final value of either $0$ or $2$.

The problem

Please read @alex.jordan's answer as he cleverly limits the range of values $Z$ can reach (say, terminal Z, or $Z_t$) to $\{-1,0,1,2\}$. As a result, the problem is now reduced to:

• For any prime number, prove that $-1$ cannot be a terminal.

Also being discussed: here

Which has been puzzling me over the last year.

The algorithm Given a natural number n define a procedure as follows:

• Generate a list of primes upto and including n
• Assign Z = n
• If Z is positive, subtract the largest prime from list which we haven't considered yet. Otherwise, add it to Z.
• Repeat until all primes have been 'considered'.

For example, take 25. The list of primes would be [2 3 5 7 11 13 17 19 23]. First prime we take is 23. Subtracting, we get Z=2. Next, we get Z = 2-19 = -17. And so on.

Consequently, Z assumes values [25 2 -17 0 13 2 -5 0 3 1].

The 'pattern'

• Beginning at 3 and every other prime thereafter, following the algorithm always (seems to) land us at 1.
• For the rest of the primes, Z has a final value of either 0 or 2.

A trivial note For the primes p not classed under 3, there is a simple explanation why they can't reach 1. It would mean, the partition of the set S containing 1 and primes till p would have an even total sum. This is clearly a contradiction.

The problem I would like to know, for instance, what mathematical tools would be best suited to attempting a proof? Perhaps all this is simply a consequence of something more powerful, like the Goldbach conjecture?

Maybe what I'm really expecting is some ingenious proof I can't understand, or a contradiction. I feel I've beaten this idea just about dead and can't go further with my understanding of Mathematics.

Which has been puzzling me over the last year.

The algorithm Given a natural number $n$ define a procedure as follows:

• Generate a list of primes upto and possibly including, $n$
• Assign $Z = n$
• If $Z > 0$, subtract the largest prime from list which we haven't considered yet. Otherwise, add it to $Z$. If $n$ is prime, it is assumed accounted for by the first step.
• Repeat until all primes have been considered.

For example, take $25$. The list of primes would be $2, 3, 5, 7, 11, 13, 17, 19, 23$. Subtracting, $23$ from $Z=25$, we get $Z=2$. Next, we get $Z=2-19= -17$. And so on.

Consequently, $Z$ assumes the values $25, 2, -17, 0, 13, 2, -5, 0, 3, 1$.

The Pattern

• Beginning at $3$ and every other prime thereafter, following the algorithm always (seems to) land us at $1$.
• For the rest of the primes, $Z$ has a final value of either $0$ or $2$.

The problem

I would like to know, for instance, what mathematical tools would be best suited to attempting a proof? Perhaps all this is simply a consequence of something more powerful, like the Goldbach conjecture?

Maybe what I'm really expecting is some ingenious proof I can't understand, or a contradiction. I feel I've beaten this idea just about dead and can't go further with my understanding of Mathematics.

Updates

Please read @alex.jordan's answer as he cleverly limits the range of values $Z$ can reach (say, terminal Z, or $Z_t$) to $\{-2,-1,0,1,2\}$. The missing piece to the puzzle seems to be a proof that no prime number can have $Z_t=-1$. This appears to be extendable to $Z_t(n)=-1$ iff n is even.