What is standard A4 paper size in pixels for a given resolution?

Is there a standard size of A4 paper in pixels?

Dimensions of A4 paper are 210mm × 297mm or 8.27in × 11.69in.

For 600dpi resolution, I calculate the paper size in pixels like below:

In imperial system:

Horizontal size = (8.27in)  x (600 pixels/in) = 4962 pixels
Vertical size   = (11.69in) x (600 pixels/in) = 7014 pixels

In metric system:

Horizontal = ((210mm) / (25.4mm/in)) x (600 pixels/in) = 4960.6299 pixels ≅ 4961 pixels
Vertical   = ((297mm) / (25.4mm/in)) x (600 pixels/in) = 7015.7480 pixels ≅ 7016 pixels

In this answer and this forum, it is said to be 4960 x 7016 pixels.

Why don't these numbers hold? What is the true dimensions of A4 paper in pixels in different resolutions?

A4 is precisely 210mm x 297mm. Your calculation in the metric system is correct. The 1 pixel difference is likely due to improper rounding, and anyhow it's less than 0.02% of the width.

A more precise determination of the A4 size in inches is: 8.2677 in x 11.6929, which would make the numbers match.

There is no "standard" size as it depends on the resolution you wish to print at, so your calculations are correct.

Any discrepancy is due to rounding in the calculations where those performing the calculations rounded the intermediate values rather than waiting until the end (as you have done).

A one or two pixel difference across 5,000 pixels isn't going to be noticeable.

The most important thing to note is that 210 / 25.4 is only approximately 8.27. To more decimal places it's 8.2677165 - how many decimal places you take in this calculation will determine how accurate your result is.

At 300 dpi you will get:

Horizontal size = (8.27in)  x (300 pixels/in) = 2481   
Vertical size   = (11.69in) x (300 pixels/in) = 3507

Horizontal size = ((210mm) / (25.4mm/in)) x (300 pixels/in) = 2480.3150 ≅ 2481
Vertical size   = ((297mm) / (25.4mm/in)) x (300 pixels/in) = 3507.8740 ≅ 3508

Which you can see also vary slightly with the calculations you use.