To install the latest version of Syncthing on Linux Mint, you should add the upstream official repository instead of using the default repository [1]. Open a terminal window and use curl to download the GPG key, then import the key with apt-key.

Here are the steps:

1. Open a terminal window.
2. Download the GPG key using the following command: curl -s https://syncthing.net/download-key | sudo apt-key add -
3. Add the Syncthing repository with the following command: echo "deb https://apt.syncthing.net/ syncthing stable" | sudo tee /etc/apt/sources.list.d/syncthing.list
4. Update the package list with the following command: sudo apt update
5. Install Syncthing with the following command: sudo apt install syncthing

After installation, Syncthing should create a launcher in your menu [2].

If you encounter issues with the GUI not opening or Syncthing-GTK not working after installation [7], it may be due to a version mismatch [8]. In this case, you should consider upgrading to a newer version of Syncthing.