What are these symbols that I have marked with blue?

[miregal]

This is a paper on fermat's last theorem

 

[The Highlander]

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This is a paper on fermat's last theorem

The "\(\displaystyle \parallel\)" symbol (as in your "\(\displaystyle \parallel\)z") is usually used to indicate parallelism, eg: \(\displaystyle AB\parallel CD\) would be read as "the line \(\displaystyle AB\) is parallel to the line \(\displaystyle CD\)" but I have never seen it used in Algebra and I have never come across the other symbol anywhere but maybe someone else will be able to enlighten us. ?
Does your text (book) not define these symbols for you? ?‍

 

[Dr.Peterson]

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This is a paper on fermat's last theorem

I located the paper here (among others), and it doesn't define the symbols. It would have been helpful to provide such contextual information.

This is not my field, and I'm not sure whether I've ever seen it before, but I find that the || symbol means "exactly divides" or "divides precisely", as explained here and here:

I haven't found an explanation of the first symbol, that looks like a poorly formed "does not divide", /|. Nor have I found any general number theory material that introduces and uses either symbol. That might be useful to have before trying to read this paper!

 

[topsquark]

As || does not appear elsewhere in the paper (and I can't even find a way to generate the /| in LaTeX) my guess is that these are typos for "divides" and "does not divide," respectively. As there are formatting typos in the paper it would not be a big surprise to find that these are as well.

-Dan

 

[pka]

As || does not appear elsewhere in the paper (and I can't even find a way to generate the /| in LaTeX) my guess is that these are typos for "divides" and "does not divide," respectively. As there are formatting typos in the paper it would not be a big surprise to find that these are as well.

@dan. The code [ imath] \| [/imath] without the space produces [imath]\|[/imath] which can be used for parallels, norms, evenly divisible among others.
But I agree with Dan that [imath]/|[/imath] is likely a typo or an atempt at [imath]\cancel{~|~}[/imath]

 

[Dr.Peterson]

As || does not appear elsewhere in the paper (and I can't even find a way to generate the /| in LaTeX) my guess is that these are typos for "divides" and "does not divide," respectively. As there are formatting typos in the paper it would not be a big surprise to find that these are as well.

-Dan

For those who didn't read my links, the first defines the "parallel" symbol clearly (though I've fixed some typos):

Let a and b be integers and n a positive integer.  Then aⁿ exactly divides b (denoted as aⁿ∥b) if aⁿ divides b but aⁿ⁺¹ does not divide b.  For example,  2⁴∥48.​

The fact that the usage in question corresponds to this definition in always having a power on the left indicates to me that this is indeed what it means.

On the other hand, I agree that the other is almost surely a typo for "does not divide", among other things because when I copy and paste it, I get

Lemma 1.5 Suppose that λ 6 |xy, λ k ||z, k ≥ 2. Then there exists a solution with λ 6 |xy, λ k−1 ||z.​

with the same "6 |" for both symbols,

​