The paper explores the nucleosynthetic signatures of Population III stars, the first generation of stars, focusing on pair-instability supernovae. It suggests that the first stars may have been very massive, around 100-300 solar masses. These stars could leave a distinct nucleosynthetic signature. The study examines helium cores in the mass range of 64 to 133 solar masses, corresponding to main-sequence star masses of approximately 140 to 260 solar masses. For helium core masses above 133 solar masses, without rotation, a black hole is formed and no nucleosynthesis is ejected. For lighter helium core masses, violent pulsations occur, leading to supernova-like mass ejection, but the star eventually produces a large iron core in hydrostatic equilibrium, which likely collapses to a black hole. This process cleanly separates the nucleosynthesis of pair-instability supernovae from other masses. Pair-instability supernovae produce a solar-like distribution of even-Z elements but are deficient in odd-Z elements. This is due to the lack of stable post-helium burning stages that set the neutron excess. Additionally, elements heavier than zinc are not produced due to the absence of s- and r-processes. The Fe/Si ratio is sensitive to the upper bound of the initial mass function (IMF). When combined with nucleosynthesis from lower mass stars, the pattern of deficient odd-Z elements persists. The study uses a detailed computational model to simulate the evolution of helium cores and their nucleosynthesis. It finds that pair-instability supernovae produce a significant amount of ⁵⁶Ni, making them the most energetic and brightest thermonuclear explosions. The nucleosynthetic yields are integrated over an IMF to compare with observed abundances in metal-deficient stars. The results show a distinctive signature with even-Z elements and a lack of odd-Z elements, which is more extreme than in lower mass stars. The paper also discusses the implications of these findings for observational tests, suggesting that the nucleosynthetic signature of pair-instability supernovae can be observed in very metal-deficient stars and damped Lyman-alpha systems. The study highlights the importance of understanding the initial mass function and the role of rotation in the evolution of these stars. The results suggest that the nucleosynthesis of Population III stars is dominated by pair-instability supernovae, with significant contributions from lower mass stars. The findings have important implications for understanding the chemical evolution of the universe and the formation of heavy elements.The paper explores the nucleosynthetic signatures of Population III stars, the first generation of stars, focusing on pair-instability supernovae. It suggests that the first stars may have been very massive, around 100-300 solar masses. These stars could leave a distinct nucleosynthetic signature. The study examines helium cores in the mass range of 64 to 133 solar masses, corresponding to main-sequence star masses of approximately 140 to 260 solar masses. For helium core masses above 133 solar masses, without rotation, a black hole is formed and no nucleosynthesis is ejected. For lighter helium core masses, violent pulsations occur, leading to supernova-like mass ejection, but the star eventually produces a large iron core in hydrostatic equilibrium, which likely collapses to a black hole. This process cleanly separates the nucleosynthesis of pair-instability supernovae from other masses. Pair-instability supernovae produce a solar-like distribution of even-Z elements but are deficient in odd-Z elements. This is due to the lack of stable post-helium burning stages that set the neutron excess. Additionally, elements heavier than zinc are not produced due to the absence of s- and r-processes. The Fe/Si ratio is sensitive to the upper bound of the initial mass function (IMF). When combined with nucleosynthesis from lower mass stars, the pattern of deficient odd-Z elements persists. The study uses a detailed computational model to simulate the evolution of helium cores and their nucleosynthesis. It finds that pair-instability supernovae produce a significant amount of ⁵⁶Ni, making them the most energetic and brightest thermonuclear explosions. The nucleosynthetic yields are integrated over an IMF to compare with observed abundances in metal-deficient stars. The results show a distinctive signature with even-Z elements and a lack of odd-Z elements, which is more extreme than in lower mass stars. The paper also discusses the implications of these findings for observational tests, suggesting that the nucleosynthetic signature of pair-instability supernovae can be observed in very metal-deficient stars and damped Lyman-alpha systems. The study highlights the importance of understanding the initial mass function and the role of rotation in the evolution of these stars. The results suggest that the nucleosynthesis of Population III stars is dominated by pair-instability supernovae, with significant contributions from lower mass stars. The findings have important implications for understanding the chemical evolution of the universe and the formation of heavy elements.