This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 • 10 18. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 • 10 26. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood (with an error that appears to be O(√ t log log t), where t is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in 1982 by Heath-Brown.

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Mit 9 Abbildungen I n h a l t s u b e r s i c h t : Es wurde die Temperaturabhiingigkeit der magnetischen Suszeptibiliet und die paramagnetische Elektronenresonanz von 12 Elbmentdipyridylen untersucht.Bei den Hauptgruppenverbindungen und ScDipy, ist der Magnetismua infolge Spinwechselwirkung von Radikalanionen stark temperaturabhgngig. In den ubrigen Ubergangsmetallverbindungen ist die Temperaturabhangigkeit weniger ausgepragt. Es wird versucht, sie auf Grund qualitativer Betrachtungen nach der LCAO-&TO-Theorie zu erklaren.A b s t r a c t : The temperature dependence of the magnetic susceptibility and the EPR were investigated on 12 element dipyridyls. The magnetism of the main group compounds and of ScDipy, depends strongly on the temperature, owing to spin interaction of radical anions. l'he temperature dependence is less marked in other transition metal compounds. It is attempted to explain the dependence by means of the LCAO-NO theory.