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The Thirteen Primes are the original holders of the title Prime and the first Transformers to be created on Cybertron in many incarnations of the franchise, having been created by writer Simon Furman.
In "Transformers: Reaching the Omega Point" and "Ask Vector Prime", it is indicated that the Thirteen Primes were proceeded by another group, the Covenant of Primus, who were created as a safeguard against Unicron rising again with no Prime to stand against him. Around the same time Primus created Primon, sometimes identified as the first bearer of the Matrix of Leadership, and later created the Thirteen Primes using Primon and the Covenant as a model after transforming himself into Cybertron. However, having never encountered the Covenant, Vector Prime was uncertain as to whether they truly existed or were simply a legend crafted by the scheming Liege Maximo, or if they-with Primon-might even be variants of the Thirteen. The Thirteen were likewise created to answer the threat of Unicron, but to do so directly while the Covenant were a last resort. 
The Thirteen are also known-in a few realities-to have inspired the gods of Greek mythology through interacting with ancient humans via holo-avatars: Prima was Zeus, Nexus was Hera, Onyx was Artemis, Vector was Hermes, Autonomous was Apollo, Maximo was Aphrodite, Megatronus was Ares, Solus was Hephaestus, Alpha Trion was Athena, and Alchemist was Dionysus, though it became unclear in other universe.  In their efforts to fight Unicron and safeguard the universe, the Thirteen created numerous weapons and tools, such as the Terminus Blade and the Star Saber. The former was intended to allow the Thirteen to travel between different dimensions, but was eventually considered obsolete due to advancing technology and carelessly cast aside. The Star Saber, by contrast, was so powerful that at least one of the Thirteen, Megatronus, began to covet it for himself, and it was eventually split into five components by Solus Prime and hidden by Nexus Prime's separate selves. The Thirteen were eventually divided as well, with some-such as Megatronus and Liege Maximo-turning their energies towards conquest.
Eventually the multiverse came under serious threat, both from the machinations of Unicron and the actions of universe-hopping villains such as the Shattered Glass version of Ultra Magnus, who had obtained the Terminus Blade. Nexus Prime's components reformed their original self and hunted down the components of the Star Saber before retrieving the Terminus Blade from the evil Autobot. Nexus then traveled to a point in the multiverse where the barriers between realities were at their weakest, and set about using the two mystic weapons to reinforce them. The other Thirteen arrived in response to his actions, some hoping to stop him while others agreed with his decision and offered their help. As a result, dimensional travel was made impossible to all but a handful, and beings like the Thirteen, Unicron, and Primus-previously singular beings existing across the entire multiverse-were separated into separate incarnations of themselves across the various realities.
Prior to this, the versions of the Thirteen had begun to blur together, at least in the mind of Vector Prime. 
According to Beast Wars: Reborn, there once existed a group of Transformers known as the Ancients-presumably an analogue of the Thirteen-of whom Vector Prime was a member. The group also included Logos Prime, who was forgotten after he turned his back on the ideals of the Ancients. He would later abduct Optimus Primal and Predacon Megatron from their home reality following their final battle at the conclusion of Beast Machines and pit them against each other in order to determine which of them was worthy of receiving his powers. Vector Prime, arriving at Primus' urging to investigate the disturbance, was forced to confront his fallen brother in a bid to stop him.
Prior to the events of Transformers: Cybertron, Vector Prime was teamed with Alpha Trion, Omega Prime, and Sentinel Maximus, the latter pair being heroic warriors from other universes, to deal with a threat posed by various minions of Unicron.
In Robots in Disguise, the Thirteen are revealed to have banished Megatronus to another dimension after his murder of Solus Prime. When Megatronus makes a bid at returning and exacting his revenge, six of the Thirteen-including Vector and Micronus Prime-begin training Optimus Prime in preparation for confronting Megatronus. After Optimus has spent a considerable amount of time in the mystical Realm of the Primes, his trainers sense Megatronus' impending escape and reluctantly dispatch Optimus to Earth, granting him portions of their own power in preparation for the conflict. With the aid of Bumblebee and his teammates on Earth Optimus apparently succeeds in destroying Megatronus, and some weeks later the Primes reclaim their power from him, leaving him drastically weakened.
Transformers: The Covenant of Primus details much of the history of the Thirteen in this continuity, including their creation, preparation to battle Unicron, the conflict itself, and their eventual breaking. It is narrated by Alpha Trion, who in this continuity is a member of the Thirteen.
The novels Transformers: Exodus, Transformers: Exiles, and Transformers: Retribution are likewise set in this continuity, but differ somewhat from The Covenant of Primus in describing the fates and relations of the Thirteen. Most notably, while in The Covenant the Liege Maximo perishes and Megatronus departs Cybertron as the remorseful Fallen, in the novels the Liege Maximo fled Cybertron and was pursued by Alchemist Prime while the Fallen departed but retained his murderous intentions. Aside from Alpha Trion, who serves in a mentor capacity to the Autobots in similar fashion to his Generation 1 incarnation, Nexus Prime and Vector Prime also appear in Exiles, where Optimus Prime encounters the former's separate forms and the fragments of the latter's Blades of Time artifact. Nexus Prime eventually forms and aids the Autobots in battling the Decepticons and Star Seekers, while Optimus finds Vector Prime in a pocket reality he created to escape the tumult of the regular universe.
Due to the different interpretations of the Transformers history, the membership of the Thirteen varies from continuity to continuity; while some members are consistent throughout the various interpretations, others are interchangeable:
- Prima-First Prime and leader
- Vector Prime-Guardian of time
- Logos Prime (Multiversal) or Alpha Trion (Aligned)-The traitorous guardian of logic or the wise mentor of Optimus Prime
- Solus Prime-Craftswoman and the only named female in all incarnations of the Thirteen, though Nexus Prime reference to "sisters" among the Thirteen in Timelines may indicate others
- Micronus Prime-First Mini-Con
- Alchemist Prime/Maccadam
- Nexus Prime-First Combiner
- Onyx Prime-First Transformer with Beast Alternate Mode
- Amalgamous Prime/Adaptus (Aligned and others) or Solomus (Multiversal-though it was hinted he and Alpha Trion might be alternate counterparts)-the Master of Forms or the Personification of Wisdom
- Quintus Prime (Aligned) or Epistemus (Multiversal)-the creator of the Quintessons or the Embodiment of Knowledge
- Liege Maximo-Founder of the Decepticons or traitorous member of the Thirteen
- Megatronus/The Fallen-Guardian of entropy and traitor to the Thirteen
- Autonomous Maximus (Multiversal) or Optimus Prime (Aligned/IDW Publishing)-the last ditch guardian of Cybertron or the most well known leader of the Autobots