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Download latest boost from:

https://www.boost.org/users/download/

https://www.boost.org/doc/libs/1_83_0/more/getting_started/unix-variants.html

move the downloaded boost_*.tar.bz2 file to a new folder (e.g “boost-ver”), change directory to there and

tar --bzip2 -xf boost_*.tar.bz2

move to the new folder boost_* and assuming you want to build boost for python3.10:
copy “user-config.jam” from tools/build/example/user-config.jam to your $HOME folder and add the line:

using python : 3.10 : /usr/bin/python3.10 : /usr/include/python3.10 : /usr/lib/python3.10 ;

set the installation folder to a local folder “./TMP” and select to install python and mpi from the above new boost_* folder:

./bootstrap.sh --prefix=./TMP --with-libraries=python,mpi

./b2 install

this should produce the boost libraries and include header in the local installation folder.
copy the local “boost” subfolder to the standard include path “/usr/local/include”

sudo cp -r ./TMP/include/boost /usr/local/include

copy the contents of the local lib subfolder to the standard lib path “/usr/lib”

sudo cp  ./TMP/lib/libboost_python310* /usr/lib/

libboost_python310.a
libboost_python310.so
libboost_python310.so.1.85.0