Are there any examples of mathematical ideas being communicated in an anonymous manner that had a substantial impact on mathematics at the time. It seems to me that it is very rare for an author to not take credit for their work, rarer still for it to have a great impact.
The t-statistic was introduced in 1908 by William Sealy Gosset, a chemist working for the Guinness brewery in Dublin, Ireland... The t-test work was submitted to and accepted in the journal Biometrika and published in 1908. Company policy at Guinness forbade its chemists from publishing their findings, so Gosset published his statistical work under the pseudonym "Student".
Of course eventually the secret came out of the bag.
Anonymity can be motivated by a variety of reasons.
The earliest example is early Pythagoreans, who had a rule of attributing all discovering to "the Man", as they called Pythagoras. As a result, we do not know the names of the discoverers of the Pythagorean theorem, incommensurability of side and diagonal of a square, etc. although it is fairly certain that it wasn't Pythagoras. And those results were the foundation of Greek geometry and formed a large portion of Euclid's Elements.
Sophie Germain corresponded first with Lagrange and then Gauss under the pseudonym Monsieur Antoine-August Le Blanc, the name of a former student she tutored. The 1804-1807 letters to Gauss presented early stages of her celebrated work on Fermat's Last Theorem, which influenced subsequent developments in the field, notably Kummer. She explained the reason when disclosing her identity to Gauss:"fearing the ridicule attached to a female scientist". Gauss's reply indicates his appraisal:"when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.
Below are lighter examples that are not too great in terms of impact, but interesting nonetheless.
When Bernoulli challenged European mathematicians to find the curve of fastest descent, the brachistochrone, one of the solutions arrived anonymously. But the solution itself was identification enough. "Ex unge leonem", Bernoulli famously wrote outing Newton, "the lion is recognized by his claws". Biographers later found out Newton's reason:"I do not love to be dunned and teased by foreigners about mathematical things..."
What we call the towers of Hanoi first saw the light of day in 1883, when Professor N. Claus (de Siam), mandarin of college Li-Sou-Stian described a version of the puzzle complete with a temple of Brahma story asserting its ancient origins. Permuting the letters of N. Claus (de Siam) Li-Sou-Stian reveals the real author's name Lucas d'Amiens of lycee Saint-Louis, of the Lucas numbers fame. Sometimes it is a hoax for publicity effect. Some online sources are still under the impression that the temple of Brahma was real.
An example whose "great impact" is debatable but which was definitely anonymous (and recent) is the anonymous 4chan poster who made significant progress on the problem of finding optimal superpermutations.
Sophie Germain had (and is still having) a great impact on mathematics and physics, but had to publish using a pseudonym Monsieur Le Blanc.