
Samsung Galaxy Note3
The latest firmware for Samsung Galaxy Note3 (SM-N900P) is N900PVPUEPH1 with Android 5.0 (Lollipop), released on July 31, 2016 for 2 regions (CSC). Every official firmware build for SM-N900P, newest first: one row per build with all regions (CSC) that received it.
Each firmware build has its own page at https://samfirm.shop/build/{PDA}, where {PDA} is the build number from the list above. Example: https://samfirm.shop/build/N900PVPUEPH1
SM-N900P started on Android 4 and has received 1 major Android upgrade; it now runs Android 5.0 (Lollipop). The newest build, N900PVPUEPH1, was released on July 31, 2016. No new build has appeared for 124 months, so regular updates for this model have most likely ended.
Specifications of SM-N900, a regional variant of the same device. Chipset and SIM lines are left out: they can differ by region.

SM-N900V

SM-N910V

SM-N915V

SM-N916S

SM-N920AZW
The newest firmware for SM-N900P runs Android 5.0 (Lollipop): build N900PVPUEPH1, released on July 31, 2016.
The latest build for SM-N900P, N900PVPUEPH1, was released on July 31, 2016 (10 years ago).
SM-N900P has had no new firmware for 124 months, so regular updates have most likely ended.
Yes, all firmware files for SM-N900P are 100% official, original, and obtained directly from Samsung servers.
To flash SM-N900P, you need to download the appropriate firmware, put your device into Download Mode, and use the Odin tool to load the BL, AP, CP, and CSC files.
Downgrading is only possible if the bootloader version (binary bit) of the old firmware matches the current one. Always check the binary version before flashing SM-N900P.