He was self-taught in his youth and must have had little patience with secondary education since he never graduated from high school. This impatience, especially with mathematical education, was to last the rest of his life.When he was eighteen years old he began to solve many of the number theory problems which were posed in the American Mathematical Monthly, regularly submitting solutions. In addition to solving problems, he began to pose problems himself. By 1902 he was contributing papers to the Monthly. For example he published two short papers in 1902 A Problem Connected with Mersenne's Numbers and Applications of a Theorem Regarding Circulants.
In 1904 he collaborated with Birkhoff on a paper on the prime factors of an - bn published in the Annals of Mathematics. In fact the result they proved was not new, although they were not aware of the earlier work which had been published by A S Bang in 1886. Also in the year 1904, Vandiver published On Some Special Arithmetic Congruences in the American Mathematical Monthly and, although still working as an agent for his father's firm, he did attend some graduate lectures at the University of Pennsylvania. He also began reading papers on algebraic number theory and embarked on a study of the work of Kummer, in particular his contributions to solving Fermat's Last Theorem. Over the next few years he published papers such as Theory of finite algebras (1912), Note on Fermat's last theorem (1914), and Symmetric functions formed by systems of elements of a finite algebra and their connection with Fermat's quotient and Bernoulli's numbers (1917).
The outbreak of World War I in 1914 did not directly affect the United States since the Democratic president Woodrow Wilson made a declaration of neutrality. This policy was controversial but popular enough to see him re-elected in 1916. However US shipping was being disrupted (and sunk) by German submarines and, under pressure from Republicans, Wilson declared war on Germany on 6 April 1917. Vandiver joined the United States Naval Reserve and continued to serve until 1919 when the war had ended. After leaving the Naval Reserve, Birkhoff persuaded Vandiver to become a professional mathematician and to accept a post at Cornell University in 1919. Despite having no formal qualifications, his excellent publication record clearly showed his high quality and he was appointed as an instructor. He also worked during the summer with Dickson at Chicago on his classic treatise History of the Theory of Numbers. In 1924 he moved to the University of Texas where he was appointed as an Associate Professor. He spent the rest of his career at the University of Texas, being promoted to full professor in 1925, then named as distinguished professor of applied mathematics and astronomy in 1947. He continued in this role until he retired in 1966 at the age of 84.
Vandiver was awarded the Cole Prize in Number Theory by the American Mathematical Society in 1931:-
... for his several papers on Fermat's last theorem published in the Transactions of the American Mathematical Society and in the Annals of Mathematics during the preceding five years, with special reference to a paper entitled "On Fermat's last theorem".In particular the paper entitled On Fermat's last theorem which was specially mentioned, was published in the Transactions of the American Mathematical Society in 1929.
His publication record was extraordinary containing 174 items throughout his career. Let us mention a few more of his early papers: The generalized Lagrange indeterminate congruence for a composite ideal modulus (1917); A property of cyclotomic integers and its relation to Fermat's last theorem (1919); A new type of criteria for the first case of Fermat's last theorem (1924); and A property of cyclotomic integers and its relation to Fermat's last theorem (1925). He continued to work on extending Kummer's methods to show that the theorem was true for increasingly large exponents. With hand calculations he, with help from his students, had shown the result to be true for all n up to 600. In 1952 he was able to implements his methods on early computers at the National Bureau of Standards Institute at Los Angeles and was able to prove the theorem true for all primes less than 2000. It is his life-long work on Fermat's Last Theorem for which he is best known, but Vandiver also wrote papers on cyclotomic fields, Bernoulli numbers, the reciprocity laws, finite fields, techniques for factorisation, semigroups, semirings, and algebras. A conjecture, now known as 'Vandiver's conjecture', concerning the class group of cyclotomic fields was so named since Vandiver frequently posed it. He was not, however, the first to make the conjecture which should really be named 'Kummer's conjecture' since it first appears in 1849 in a letter which Kummer wrote to Kronecker.
Although the Cole Prize might be considered Vandiver's greatest distinction, we should also mention that he was vice-president of the American Mathematical Society in 1934-35 and was the Colloquium Lecturer at Ann Arbor in 1935 when he lectured on Fermat's Last Theorem. He was also honoured with election to the National Academy of Sciences in 1934 and he was given an honorary degree by the University of Pennsylvania in 1946.
Vandiver married Maude Folmsbee in 1923; they had one son, Frank Vandiver, who became president of Texas A&M. It is worth noting that Vandiver continued to hold strong views against public education and Frank was privately tutored to first degree level without attending high school or university. Vandiver never owned a house and lived with his wife in the Alamo Hotel in Austin where he had a large collection of classical recordings. D H Lehmer writes :-
Vandiver's best method of communicating ideas was that of personal face to face discussion; he was a poor lecturer. In these discussions he conveyed not only the mathematical ideas, but also his ideas about mathematics. These were often strong and well put and reminded one of his contemporary, E T Bell. During such a session it was not unusual to have him inject his opinions about other topics such as baseball and Mozart. He was a man of many parts with a great mind of his own.
Article by: J J O'Connor and E F Robertson